Algorithmic Lie theory for solving ordinary differential equations

Algorithmic Lie Theory for Solving Ordinary Differential Equations

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Algorithmic Lie Theory for Solving Ordinary Differential Equations

Fritz Schwarz Fraunhofer Gesellschaft Sankt Augustin, Germany

Boca Raton London New York

Chapman & Hall/CRC is an imprint of the Taylor & Francis Group, an informa business

Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑1‑58488‑889‑5 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright. com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Schwarz, Fritz, 1941‑ Alogrithmic lie theory for solving ordinary differential equations / Fritz Schwarz. p. cm. Includes bibliographical references and index. ISBN 978‑1‑58488‑889‑5 (alk. paper) 1. Differential equations‑‑Numerical solutions. 2. Lie algebras. I. Title. QA371.S387 2007 518’.63‑‑dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

2007020633

To my Mother

Contents 1 Introduction

1

2 Linear Differential Equations 2.1 Linear Ordinary Differential Equations . . 2.2 Janet’s Algorithm . . . . . . . . . . . . . . 2.3 Properties of Janet Bases . . . . . . . . . . 2.4 Solving Partial Differential Equations . . .

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11 11 40 55 77

Transformation Groups Lie Groups and Transformation Groups . . . Algebraic Properties of Vector Fields . . . . Group Actions in the Plane . . . . . . . . . . Classification of Lie Algebras and Lie Groups Lie Systems . . . . . . . . . . . . . . . . . .

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105 105 115 121 135 143

4 Equivalence and Invariants of Differential Equations 4.1 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Nonlinear First Order Equations . . . . . . . . . . . . . . . . 4.3 Nonlinear Equations of Second and Higher Order . . . . . .

159 162 176 186

5 Symmetries of Differential Equations 5.1 Transformation of Differential Equations . . . . 5.2 Symmetries of First Order Equations . . . . . . 5.3 Symmetries of Second Order Equations . . . . . 5.4 Symmetries of Nonlinear Third Order Equations 5.5 Symmetries of Linearizable Equations . . . . . .

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193 194 206 210 224 236

6 Transformation to Canonical Form 6.1 First Order Equations . . . . . . . . 6.2 Second Order Equations . . . . . . 6.3 Nonlinear Third Order Equations . 6.4 Linearizable Third Order Equations

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247 247 249 279 298

7 Solution Algorithms 7.1 First Order Equations . . . . . . . . 7.2 Second Order Equations . . . . . . 7.3 Nonlinear Equations of Third Order 7.4 Linearizable Third Order Equations

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311 312 319 328 345

3 Lie 3.1 3.2 3.3 3.4 3.5

ix

x 8 Concluding Remarks

351

A Solutions to Selected Problems

355

B Collection of Useful Formulas

377

C Algebra of Monomials

383

D Loewy Decompositions of Kamke’s Collection

387

E Symmetries of Kamke’s Collection

403

F ALLTYPES Userinterface

417

References

419

Index

431

Chapter 1 Introduction

“...Es geht mir vorw¨ arts, aber langsam, und es kostet unendliche und ¨ außerst langweilige Rechnungen...”1 Sophus Lie, Letter to A. Mayer of April 4, 1874.

Solving equations has been one of the most important driving forces in the history of mathematics. Particularly well known is the problem of solving algebraic equations of order higher than four and its eventual solution by Lagrange, Ruffini, Abel and Galois. The complete answer given by the latter author provided not only the solution for the problem at hand, but also established a new field in mathematics, the theory of groups, which in turn laid the foundation of modern algebra. A knowledgeable and detailed review of this subject and various other topics discussed below may be found in the book by Wussing [190], see also the Historical Remarks by Bourbaki [16]. Much less known is the fact that the theory of differential equations took a similar course in the second half of the 19th century. At that time, solving ordinary differential equations (ode’s) had become one of the most important problems in applied mathematics, about 200 years after Leibniz and Newton introduced the concept of the derivative and the integral of a function. Numerous phenomena in the physical sciences were described by formulas involving differentiation and integration, and the need arose to determine the functional dependencies between the variables involved. In other words, the problem of solving a differential equation was born. In this book the phrase solution of an ode means an expression for the general solution in some function field, e.g., in terms of elementary or Liouvillian functions. The differential equation should vanish identically upon its substitution. If n is the order of the equation the general solution involving n constant parameters is searched for. It should not involve derivatives, infinite series or products. Furthermore, any numerical or graphical representation is excluded.

1 Translation

by the author: “...I proceed, but slowly, and it takes me infinite, extremely boring calculations...”

1

2 In the course of time, several ad hoc integration methods had been developed for special classes of ode’s which occurred in the description of practical problems. It was quickly realized that a common feature of the various solution procedures consists in introducing new variables such that the given equation is transformed into an equation with known solution, or for which an integration method is available. Due to its importance, the special term equivalence problem was created for the task of recognizing whether two equations may be transformed into each other, and the entirety of equations sharing this property is called an equivalence class. In this general form, however, it is hardly justified to speak of a solution method because there is no indication of how to discover such a transformation in a concrete case, although infinitely many may exist. In particular, for nonlinear equations, efforts to find a solution along these lines very often terminate in the worst case possible: The attempts are abandoned without any evidence whether or not a solution in closed form does exist. It is therefore highly desirable to develop algorithmic solution schemes for the largest possible classes of equations. The importance of equivalence problems had been realized before in algebraic geometry where the behavior of algebraic forms under the action of certain transformation groups, and in particular its invariants w.r.t. to these transformations, had been studied. The analogy with these methods turned out to be extremely successful in the realm of differential equations. In his article on invariants, Forsyth [47] expresses this as follows: Similarity in properties of differential equations and of algebraic equations has long been of great value, both in the development of the theory and in the indication of methods of practical solution of the former equations. Based on preceding work by Cockle [31], Laguerre [99], Brioschi [19] and Halphen [64], Forsyth applied this principle in particular for calculating invariants of linear differential equations under predefined classes of transformations. This was essentially the state of affairs when Sophus Lie got interested in this problem in the sixties of the 19th century. The most important guide for him was Galois’ theory for solving algebraic equations that had become widely known due to Liouville’s efforts around 1850. The most significant recognition of Sophus Lie was that the transformation properties of an ode under certain groups of continuous transformations play a fundamental role in answering this question, very much like the permutations of the solutions of an algebraic equation furnish the key to understanding its solution behavior. This is best explained by his own words, which are taken from the first chapter of his book [113] on differential equations: 2 2 Translation

by the author: “Previous investigations on ordinary differential equations as they may be found in the customary textbooks do not form a systematic entirety. Special integration theories have been developed, e.g., for homogeneous differential equations, for linear differential equations and other special forms of integrable differential equations. The mathematicians failed to observe, however, that the special theories may be subordinated to a general method. The foundation of this method is the concept of an infinitesimal transformation and closely related to it the concept of a one-parameter group.”

Introduction

3

“Die ¨ alteren Untersuchungen u ¨ber gew¨ ohnliche Differentialgleichungen, wie man sie in den gebr¨ auchlichen Lehrb¨ uchern findet, bilden kein systematisches Ganzes. Man entwickelte specielle Integrationstheorien z. B. f¨ ur homogene Differentialgleichungen, f¨ ur die linearen Differentialgleichungen und andere specielle Formen von Differentialgleichungen. Es war aber den Mathematikern entgangen, dass diese speciellen Theorien sich unter eine allgemeine Methode unterordnen lassen. Das Fundament dieser Methode ist der Begriff der infinitesimalen Transformation und der damit aufs engste zusammenh¨ angende Begriff der eingliedrigen Gruppe.”

In order to provide an idea of how Lie acquired the intuition that finally led him to create his theory, and to understand the basic principle underlying it, a few simple but typical examples will be presented. Consider first the equation y 00 (y + x) + y 0 (y 0 − 1) = 0 (1.1) from the collection of solved equations by Kamke [85], equation 6.133 in his enumeration. If new variables u and v are introduced by x = u + v, y = u − v with v ≡ v(u), equation (1.1) assumes the form 2v 00 u + v 02 − 1 = 0. On the other hand, the transformation x = u + 2v, y = 3u − 4v yields v 00 u − 21 v 00 v + 52 v 03 −

9 02 10 v

+

1 0 20 v

+

3 10

= 0.

Finally x = 21 (v 2 + u), y = 12 (v 2 − u) gives the particularly simple equation v 00 v 3 +

1 4

= 0.

(1.2)

From this latter equation the solution is readily obtained in the form C1 v 2 = C12 (u + C2 )2 −

1 4

by means of two quadratures as may easily be verified by substitution. 3 Thus finding among the infinity of possible variable transformations the particular one yielding (1.2) amounts to solving equation (1.1). Equation (1.2) has another remarkable property. Any of the following substitutions to new variables z and w ≡ w(z), u = −1 − z1 , v = − w z,

+4 w u = − 4z 8z + 7 , v = − 8z + 7 or

w −9 u = − 9z z − 10 , v = − z − 10 ,

(1.3)

does not change its form, i.e., the result in all three cases is w00 w3 + 14 = 0. A transformation with this particular property is called a symmetry of the 3 The

details of this example may be found on page 323.

4 differential equation. The three substitutions (1.3) may be subsumed under the class of transformations of the form u=

(z + a)b2 , 1 − (z + a)cb2

v=

wb 1 − (z + a)cb2

(1.4)

corresponding to the parameter values (a, b, c) = (1, 1, 1), (1, 2, 2) and (−1, 3, 1) respectively. Most remarkable, the entirety of transformations (1.4) forms a group whose elements are parametrized in terms of a, b and c. It turns out that invariance of (1.2) under the group of transformations (1.4) is a characteristic feature not only of eq. (1.2) but of all equations obtained from it by a variable transformation, i.e., for the full equivalence class determined by (1.2) which may be taken as a canonical representative. In general, a canonical form is not uniquely determined by the symmetry type. The allowed freedom may be used in order to obtain an especially simple solution. An invariance transformation of any member of the equivalence class determined by (1.2) is obtained from (1.3) or (1.4) if the same variable transformation is applied to it that leads to the transformed equation. The first transformation of (1.3) for example yields the symmetry x=

v 1 2 − 2, (u − v)

y=

u 1 2 + 2 (u − v)

of (1.1). The entirety of invariance transformations related by a variable change is called a symmetry type. In general a group of transformations may be a symmetry group for equations that are not equivalent to each other, e.g., if they have not the same order. The entirety of equations allowing the same symmetry type is called a symmetry class, it is the union of equivalence classes. With the insight gained by examples similar to those just described, Lie developed a solution scheme for ode’s that may be traced back to the following question: Are there any nontrivial transformations leaving the form of the given equation invariant? If the answer is affirmative like for eq. (1.2) above, they determine a canonical form of the given equation. If the solution of the canonical equation may be obtained and moreover it is possible to determine the transformation of the given equation to a canonical form, the original problem is solved. Due to the crucial role of various kinds of transformations in Lie’s theory of differential equations, a major part of his work is devoted to establishing a systematic theory of groups of continuous transformations. Although it was only a device for his main goal of solving ode’s, the theory of Lie groups and the closely related Lie algebras as they have been named by Hermann Weyl became well-established fields in mathematics, independent from differential equations. Contrary to that, Lie’s theory for solving ordinary differential equations has been almost forgotten after his death. Most textbooks on differential equations do not even mention his name or the concept of a symmetry. For a long

Introduction

5

time the practical work of solving ode’s has remained almost unchanged from the way it is described in the above quotation by Lie. Basically it consists of a set of heuristics that are applied one after another, usually accompanied by database like tools in the form of collections of solved examples as the one by Kamke [85], or the more recent collections by Polyanin [149] or Sachdev [157]. This fact certainly needs some explanation. The most important obstacle for applying his methods is the fact that they require huge amounts of analytical calculations that hardly may be performed by pencil and paper for almost any nontrivial example. The obvious remedy is to employ one of the computer algebra systems that have become available lately to carry them through. There is another more fundamental reason. It is related to the question to what extent is his theory constructive, i.e., how much effort is needed to design algorithms from it that may be applied for solving problems? This is particularly important if the goal is a piece of computer algebra software that accepts an ode as input and returns the best possible answer according to the underlying theory. Lie himself did not attach a high value to this goal because he probably realized the practical difficulties due to the size of calculations. After Lie had recognized that the symmetry of an ode is a fundamental concept that allows finding its solutions in closed form, he described several solution procedures based on it. On the one hand, if the infinitesimal generators for the symmetry group of a given ode may be determined explicitly, the canonical form and possibly the solution may be constructed from it. This is described in detail in Lie’s book [113] on the subject; see also the books by Olver [140], Stephani [174] and Bluman [13]. In a second version the vector fields generating the symmetry transformations are also determined explicitly. The proper integration problem is reformulated and solved in terms of an associated linear partial differential equation (pde), a subject that Lie had considered in detail about ten years earlier. This procedure is also described in Lie’s book quoted above. There is still another approach of solving equations with symmetries. Its symmetry type is determined by suitable manipuations of the determining equations of the symmetries without solving them explicitly. It is shown that in principle the transformations to canonical form may be obtained from it. Lie [109] outlined this proceeding in a series of articles in the Archiv for Mathematik but never came back to it. The reason probably is that it does not seem possible to set up the equations for the finite transformations in terms of the given ode to be solved. In order to make this latter approach into a working method, two fundamental discoveries that were made within about twenty years after Lie’s death turned out to be of fundamental importance. The first is Loewy’s theory of linear ode’s, Loewy [127], and its generalization to certain systems of linear pde’s [105]. This was another effort to use the analogies between algebraic equations and differential equations, i.e., the reducibility of a linear ode and its representation in terms of irreducible equations of lowest degree. Loewy’s main result is a theorem that guarantees the existence of a unique decomposition in terms of largest so-called completely reducible factors. The second is

6 Janet’s theory of linear pde’s, Janet [83], and in particular the canonical form he introduced for any such system which is known today as a Janet basis. This concept is closely related to Buchberger’s Gr¨ obner basis in polynomial ideal theory, and turned out to be fundamental in the realm of differential equations. The importance of these results for Lie’s theory originates from the fact that the symmetries of any differential equation are obtained from its so-called determining system, a linear homogeneous system of pde’s. Representing it as a Janet basis allows one to identify its symmetry type from the coefficients. Furthermore, Loewy’s and Janet’s results allow to break down the solution algorithms for nonlinear ode’s based on its symmetries uniquely and systematically into a small number of basic problems for which solution algorithms may be designed. At each step and in particular for the final result, the function field necessary for representing the answer is obtained. Sophus Lie was not the only person who had the idea of generalizing Galois’ results to differential equations. Almost at the same time as he developed his theories in Leipzig, Picard and Vessiot [146] in Paris initiated another effort with a similar objective. Their approach is usually referred to as PicardVessiot-theory or differential Galois theory. Their goal is more limited than Lie’s because they consider only linear ode’s. Very much like ordinary Galois theory deals with the field extensions generated by the solutions of univariate algebraic equations, differential Galois theory investigates the differential field extensions over a base field generated by the solutions of linear ode’s. They obtain a theory that establishes a one-to-one correspondence between the structure of the differential Galois group of an equation and the function field generated by its solutions. Therefore the theory of Picard and Vessiot may very well claim to be the proper analogue of Galois’ theory for linear ode’s. Although Lie aimed for a general and complete theory for solving nonlinear ode’s as well, this goal has only partially been achieved. In Drach’s [38] thesis ´ Picard and J. Tannery, the state of affairs is that has been supervised by E. expressed as follows: ...Lie’s th´eorie des groupes ` a l’int´egration des ´equations n’est pas la v´eritable g´en´eralization de la m´ethode employ´ee par Galois pour les ´equations alg´ebriques.4 Drach’s main objections are that Lie’s theory is not complete in the following sense: An equation may have a symmetry group that cannot be employed for solving it. This holds for first order equations, and may occur for equations of higher order as well. On the other hand, Lie’s theorems are not reciprocal, i.e., an equation may have a closed form solution yet may not have a nontrivial symmetry group. An example is the equation given by Gonzales-Lopez [54] y 00 y − y 02 − y 0 y 2 (x2 + 1) − 2y 3 x = 0.

4 Translation

(1.5)

by the author: “...Lie’s group theory for the integration of equations is not the true generalization of the method applied by Galois for algebraic equations.”

Introduction

7

This equation has a trivial symmetry group, yet its general solution is y=

C13 . (C1 x + 1)2 + C12 + 1 + C2 eC1 x

(1.6)

In other words, Lie’s theory does not establish a unique relation between the existence of symmetries of an equation and its solution behavior. In this context the following question is frequently asked: What is the relationship between Lie’s theory and the theory of Picard and Vessiot? Although their point of departure is basically the same and group theory is an essential part of both theories, the answer is that there is virtually no connection between them. They are different approaches to the general problem of solving ordinary differential equations in closed form. Consequently, there arises a second question: What is the proper generalization of Galois’ theory for solving nonlinear differential equations? There does not seem to exist a satisfactory answer at present. Another important aspect is raised by the following question: To what extent do these theories support solving concrete problems? Although the theory of Picard and Vessiot is quite satisfactory from a theoretical point of view, it is only of limited value for obtaining explicit solutions. On the other hand, as it will turn out later on in this book, for solving nonlinear ordinary differential equations Lie’s theory is a powerful tool. The solutions for the nonlinear second order equations in Chapter 6 of Kamke’s collection, for example, are almost exclusively due to Lie symmetries. Furthermore, any equation obtained from them by a variable transformation may also be solved by applying Lie’s theory, i.e., instead of a finite number of examples their full equivalence classes become amenable to the solution procedure. Sometimes the objection is raised that most equations do not have nontrivial Lie symmetries and therefore Lie’s theory does not apply. Although this is true, one should keep in mind that a similar argument applies to linear ode’s and their differential Galois groups, and is also true for algebraic equations. The equations that actually occur in applications very often have a special structure that expresses itself in the existence of symmetries. It is of utmost importance to recognize these situations because it leads to a better understanding of the underlying problem, and to take advantage of them for the solution procedure. In this way Lie’s theory may finally lead to a better understanding of the distinctive features that make an equation solvable, and may even help to raise the intuition that is necessary in order to establish a more complete theory. As mentioned before, the proceeding described in Lie [109] is the basis for most of the solution algorithms for second and third order ode’s in this book. In this approach, the symmetry type of an ode, being an invariant under point transformations, serves the purpose of narrowing down the possible equivalence classes to which it may belong. The totality of equations from a certain family sharing the same symmetry type is called a symmetry class; it

8 is a union of equivalence classes. In this form the close relation between the symmetry analysis and equivalence problems becomes particularly obvious. This point has been discussed in the preface of the book by Olver [141]. After the symmetry class of an equation and a possible canonical form are known, the function field of admitted equivalence transformations must be specified in order to obtain a well-defined problem. Furthermore, it should be possible to solve equivalence problems within this function field constructively and to design solution algorithms. In this book most ode’s are polynomial in the derivatives with coefficients that are rational functions of the dependent and the independent variables. The smallest field containing the coefficients is called the base field. More general function fields are extensions of the base field, e.g., elementary or Liouvillian extensions. The overall scheme for solving second- or higher order ode’s along these lines decomposes in three major steps.  Determine the symmetry class of the given equation. This is achieved in terms of the Janet basis coefficients for the determining system of its symmetries; it proceeds completely in the base field.  Transform the equation to a canonical form corresponding to its equivalence class which is contained in the symmetry class. To this end a system of linear or Riccati-like pde’s will be set up for which solutions in well-defined function fields may be determined algorithmically.  Solve the canonical form and determine the solution of the given equation from it. In addition to developing solution schemes for individual equations, equations containing undetermined elements like parameters or functions depending on a specified set of indeterminates may be considered. Examples are linear equations, Riccati equations or Abel equations with coefficients that are undetermined functions of the independent variable. In other words, whole families of equations are considered that are identified by a certain structure. The problem then is to decompose the totality of equations with a given structure into symmetry or equivalence classes. The importance of this proceeding arises from the fact that many equations occuring in practical problems are specified in terms of its structure. The scope of this book has been restricted deliberately to ordinary differential equations of low order. The aim is to obtain solution algorithms for welldefined classes of equations, leading to the best result including the possible answer that a solution of a certain type does not exist. Any extension of this scope would lead to a significant increase of the size of the book. The simplest pde’s for a scalar function depending on two variables, or the simplest systems of ode’s for two functions, require a classification of the finite Lie groups of three-space comprising several hundred entries. It remains an interesting

Introduction

9

and challenging problem to develop solution schemes for these cases along the lines described in this book. The goals of this book are twofold. On the one hand, it intends to be a source of reference for people working actively in the field. To this end, the detailed discussion of many subjects and the extensive tabulations in the main text and especially in the appendices should prove to be an indispensible tool. Beyond that it might serve as a textbook if supplemented by some additional reading material, e.g., on the theory of ode’s in general or on Lie groups and Lie algebras. Some bibliographic hints are given in the introductory remarks to the individual chapters. The subsequent Chapter 2 provides the mathematical foundation for the rest of the book. It deals mainly with linear differential equations. For linear ode’s the most important topic dealt with is Loewy’s theory for the unique decomposition into completely reducible components. This is the basis for handling more general problems later on. For linear pde’s a fairly complete description of Janet’s work is given. Of particular importance are those systems of pde’s that allow a finite-dimensional solution space. The dimension of this space is called the order of the system. This generalizes the notion of order of a linear ode. For values up to three, a complete classification of Janet bases is given. Loewy’s decomposition of linear ode’s is generalized to such systems. In Chapter 3, those results from the theory of continuous groups of a two-dimensional manifold R2 are presented that are relevant for the main subject of this book, i.e., solving ode’s. The close relation between Lie’s symmetry analysis and the equivalence problem is the subject of Chapter 4. The two subsequent Chapters are the core of the book. The subject of Chapter 5 is to identify the symmetry class to which a given quasilinear equation of order two or three belongs. To this end it is sufficient to know a Janet basis for its determining system, it is not necessary to determine the symmetry generators explicitly. The same is true for solving a given ode as is shown in Chapter 6. The crucial problem of transforming a given ode with a nontrivial symmetry to canonical form is achieved by solving a system of linear or Riccati-like pde’s. As mentioned above, the symmetry class serves essentially the purpose of identifying a canonical form. Lie’s second approach, i.e., to determine the Lie algebra of symmetry generators explicitly and to generate the canonical form of the differential equations from it, is the subject of Section 3 in Chapter 6. Various topics are postponed to the appendices A through F. The solutions of selected exercises are given in Appendix A. A collection of useful formulae is listed in Appendix B. Various properties of ideals of monomials that are needed for Janet’s algorithm are presented in Appendix C. In Appendix D for selected linear ode’s from Kamke’s collection the Loewy decomposition is given explicitly. In Appendix E the symmetries of the equations of second and third order of Kamke’s collection are listed. They serve as a reference in many places of this book. Appendix F contains a description of the user interface of the software system ALLTYPES which is an important part of this book.

10 It provides a great many functions for carrying out the calculations that are necessary in order to apply the theory to practical problems. Most of these calculations are too voluminous to be performed by pencil and paper. This software may be accessed through the website www.alltypes.de. The most important sources of information for this book were Lie’s original writings on the subject, edited in seven volumes of the Gesammelte Abhandlungen by Friedrich Engel, and also Engels’ Anmerkungen in the supplements to these volumes. Furthermore, there are the books written under Lie’s guidance as quoted in the references at the end of this book. A lot of interesting background information may be found in the biography of Sophus Lie by Stubhaug [175]. The author would like to thank several colleagues and students for their helpful comments. In particular, G¨ unter Czichowski, the late Eckehart Hotzel and several anonymous referees. The abiding encouragement by Carl-Adam Petri and the excellent working environment at the Fraunhofer Institute of Ulrich Trottenberg are also gratefully acknowledged.

Chapter 2 Linear Differential Equations

This chapter provides the fundamental algorithms for working with linear ordinary and partial differential equations that are the building blocks for the solution algorithms given later in this book. Systems of linear pde’s are characterized in the first place by the number m of dependent and the number n of independent variables. Additional quantities of interest are the number N of equations, the order of the highest derivatives that may occur and the smallest function field in which the coefficients are contained, called the base field. Without further specification this is assumed to be the field of rational functions in the independent variables with rational number coefficients. The special case m = n = N = 1 corresponds to a single linear ode. The fundamental concepts described in this chapter are the Loewy decomposition, Loewy [127], and the Janet basis, Janet [83]. The latter term is chosen to honour the French mathematician Maurice Janet who described this concept and gave an algorithm to obtain it. After it had been forgotten for about fifty years, it was rediscovered [163] and utilized in various applications as described in this book. A good survey is also given in the article by Plesken and Robertz [148]. Janet bases are the differential counterpart of Gr¨obner bases that have been introduced by Bruno Buchberger and are a well-established tool in polynomial ideal theory and algebraic geometry now. The relevance of a Janet basis for the main subject of this book, i.e., solving ordinary differential equations or ode’s for short, originates from the fact that the symmetries of any such equation are determined by a system of linear homogeneous pde’s. General references for this chapter are the book by Ince [80] or the two volumes on linear equations by Schlesinger [160]. For Chapter 2.1 the first 150 pages of the book by van der Put and Singer [185], or the article by Buium and Cassidy [23] are highly recommended.

2.1

Linear Ordinary Differential Equations

The theory of linear ode’s is important for several reasons. In the first place, many problems in connection with pde’s may be traced back to certain standard problems involving ode’s. Secondly, the theory of linear ode’s is

11

12 more complete than the theory of pde’s and therefore serves as a guide for what kind of results may be expected for the latter. Many of the concepts and the methods introduced in this section will be needed later on. Symmetry properties of linear ode’s will be discussed in detail in Chapter 5.5. Solving Linear ODE’s. A linear homogeneous ode for a differential indy (n−1) dy and more generally y (n) = is determinate y, for which y 0 = dx dx defined for any natural number n, will be written as L(y) ≡ y (n) + q1 y (n−1) + q2 y (n−2) + . . . + qn−1 y 0 + qn y = 0.

(2.1)

If not specified otherwise the coefficients qk , k = 1, . . . , n, are rational functions in the independent variable x with rational number coefficients, i.e., qk ∈ Q(x) which is called the base field. Equation (2.1) allows the trivial solution y ≡ 0. The general solution contains n constants C1 , . . . , Cn . Due to its linearity, these constants appear in the form y = C1 y1 + . . . + Cn yn with the yk linearly independent over the field of constants, e. g. Ci ∈ Q in this case. They form a so-called fundamental system and generate a n-dimensional vector space over the field of constants Q. The coefficients qk of (2.1) may be rationally expressed in terms of a fundamental system and its derivatives. To this end the Wronskian y1 y2 . . . yn y10 y20 . . . yn0 W (n) (y1 , . . . , yn ) = . . . (2.2) ... ... (n−1) (n−1) (n−1) y y2 . . . yn 1 is defined. It is different from zero iff the yk (x) are independent over the constants. More generally, the determinants y y1 y2 . . . y n 0 y y10 y20 . . . yn0 ∂ (n) Wk (y1 , . . . , yn ) = (2.3) (n−k) . . . . . . . . . . . . ∂y y (n) y (n) y (n) . . . yn(n) 1 2 (n)

are defined for k = 0, 1, . . . , n. There are the obvious relations W0 = (n) (n) (−1)n−1 W (n) and dW = (−1)n−1 W1 . The determinant at the right dx hand side of (2.3) vanishes for y = yk , k = 1, . . . , n. Therefore, given a fundamental system y1 , . . . , yn , (2.3) is another way of writing the left hand side of (2.1). This yields the representation (n)

qk = (−1)k

Wk (y1 , . . . , yn ) W (n) (y1 , . . . , yn )

(2.4)

Linear Differential Equations

13

for the coefficients qk , k = 1, . . . , n in terms of a fundamental system, i.e., (2.1) is uniquely determined by its solution space. For k = 1 there follows R 0 (n) W1 + q1 W (n) = W (n) + q1 W (n) = 0, i.e., W (n) = exp (− q1 dx). It is always possible to replace equation (2.1) by another equation over the same base field, its so called rational normal form y¯(n) + q¯2 y¯(n−2) + . . . + q¯n−1 y¯0 + q¯n y¯ = 0, 1 R q (x)dx
y¯. by introducing a new dependent variable y¯(x) by y = exp − n 1 More general transformations with the same effect are discussed in Chapter 4.1. Solving an equation (2.1) amounts to determining n independent elements of a fundamental system, called a full set of solutions. In order to make this a rigorous concept, the function field containing the yk ’s has to be specified. After that an algorithm has to be designed in order to obtain the solution explicitly. The simplest problem is to ask for solutions in the base field. If it is Q(x), this amounts to searching for the rational solutions of (2.1). In general any rational solution has the form y = C1 y1 + . . . + Ck yk

(2.5)

with 0 ≤ k ≤ n, yk ∈ Q(x), Ck constant and y1 , . . . , yk linearly independent over constants. It may occur that all elements of a fundamental system are rational, i.e., k = n, or that the equation does not have a rational solution at all. The most general rational solution is the one with the highest possible value for k. If k < n it is an important problem to find the minimal extension of the base field containing a full set of solutions. Determining the most general solution in the base field occurs as a subproblem of many other problems. It is therefore important that an efficient algorithm is available for this purpose. Such an algorithm is designed according to the following general principles that are valid for more general function fields as well. At first the problem has to be made finite, i.e., the possible solutions have to be parametrized such that a finite number of candidates is obtained. Secondly, a search procedure must be provided that identifies the desired answer among the candidates, or assures that a solution does not exist. The following observation yields an important constraint for the possible form of any rational solution of (2.1). Assume a rational solution has a pole of order k at the finite position x = x0 . If it is substituted into (2.1), a term 1 proportional to is generated from the highest derivative y (n) . (x − x0 )k+n Furthermore, it is assumed that both the coefficients and the solution are uniquely represented as partial fractions. Then in order to make the left hand side vanish, at least one more term of order k + n must occur. Because the lower derivatives of y cannot produce such a term, one of the coefficients must

14 contain a pole at x0 . As a consequence, the only possible poles of a rational solution of (2.1) are those that occur in the coefficients qi (x). Alternatively, if (2.1) is considered as an equation over Q[x], by multiplication with the least common multiple of the coefficient denominators, the possible poles are the zeros of the coefficient of the highest derivative y (n) . For any of these poles the coefficient of the highest order term must vanish. This leads to a nonlinear equation for the possible highest order whose largest solution in natural numbers is the desired bound. Similar arguments apply for the behavior at infinity. A polynomial ansatz with undetermined coefficients within these limits leads to a system of linear algebraic equations the solutions of which determine the rational solutions. The following algorithm is based on these considerations. More details may be found in Schwarz [162] and the references given there. Algorithm 2.1 neous ode

RationalSolutionsLode(L(y)). Given a linear homoge-

L(y) ≡ y (n) + q1 y (n−1) + q2 y (n−2) + . . . + qn−1 y 0 + qn y = 0 with qk (x) ∈ Q(x), the maximal number of linearly independent rational solutions is returned. S1 : Singularities and bounds. Identify the possible positions of poles and determine an upper bound for any of them. S2 : Solve linear system. Set up the linear system for the undetermined coefficients for an ansatz within the limits found in S1 and determine its solution. If only the trivial solution exists, return an empty list {}. S3 : Return result. For each solution found in S2 generate a solution of L(y) and return the list {y1 , . . . , yk }. 1
y 0 − 4 y = 0. It has Example 2.1 Consider the equation y 00 + 2 + x x2 no. 2.201 in Kamke’s collection. For the order N of a singularity at infinity there follows immediately N = 0. For the order M of the only other pole at x = 0 there follows M (M + 1) − M − 4 = 0, i.e., M = 2. In step S2 the ansatz a22 + ax1 + a0 yields the system 3a1 + 4a2 = 0, 2a0 + a1 = 0 with x the solution a1 = − 43 a2 , a0 = 23 a2 . With the normalization a2 = 1 the only 4 + 2 , i.e., k = 1 in rational solution obtained in step S3 has the form 12 − 3x 3 x this case. The second element of a fundamental system is not rational, it will be obtained in Example 2.14 below by a different method. Solving a linear inhomogeneous equation y (n) + q1 y (n−1) + q2 y (n−2) + . . . + qn−1 y 0 + qn y = r

(2.6)

may be traced back to the corresponding homogeneous problem with r = 0 and integrations by the method of variation of constants. To this end, the Ci in the general solution y = C1 y1 + . . . + Cn yn of the homogeneous equation

Linear Differential Equations

15

are considered as functions of x. Substituting this expression into (2.6) and imposing the constraints (k)

(k)

C10 y1 + C20 y2 + . . . + Cn0 yn(k) = 0

(2.7)

for k = 0, . . . , n − 2 yields the additional condition (n−1)

C10 y1

(n−1)

+ C20 y2

+ . . . + Cn0 yn(n−1) = r.

(2.8)

The linear system (2.7), (2.8) for the Ck0 has always a solution due to the nonvanishing determinant of its coefficient matrix which is the Wronskian W (n) . Consequently, solving the inhomogeneous equation (2.6) requires only integrations if a fundamental system for the corresponding homogeneous problem is known. Example 2.2 If y1 and y2 is a fundamental system for y 00 + q1 y 0 + q2 y = 0, the general solution of y 00 + q1 y 0 + q2 y = r may be written as Z y = C1 y1 + C2 y2 + y1

ry2 dx − y2 W

Z

ry1 dx W

(2.9)

where W = y10 y2 − y20 y1 . Let the equation y 00 −

4 e−2x 4x y0 + y= 2x − 1 2x − 1 2x − 1

be given. A fundamental system for the homogeneous equation is y1 = x, e−2x and W = (2x − 1)e2x . Substituting this into y2 = e2x . Therefore r = 2x −1 the above equation leads to Z Z xe−4x e−2x 2x 2x y = C1 e + C2 x + e dx. 2 dx − x (2x − 1) (2x − 1)2 Inhomogeneous problems will occur frequently later on if a linear homogeneous equation of third order is reducible but not completely reducible. If a linear ode (2.1) does not have a full set of solutions in its base field F, i.e., if k < n in (2.5), the dimension of its solution space may increase if an enlarged function field is admitted for a fundamental system. This is achieved by adjoining solutions of algebraic or differential equations of order not higher than n to the base field. If θ is the element to be adjoined, the enlarged differential extension field is denoted by Fhθi, extensions by more than a single element are denoted by Fhθ1 , . . . , θk i. Frequently the newly adjoined elements are required to obey special equations over the previously defined field. If these equations are algebraic or of first order, a special term is introduced for the corresponding extension.

16 Definition 2.1 (Liouvillian extensions). Let F be a differential field with derivation d ≡ 0 . A simple Liouvillian extension Fhθi is obtained by dx adjoining an element θ such that one of the following holds. i) Algebraic extension, θ satisfies a polynomial equation with coefficients in F. ii) Extension by the exponential of an integral, θ0 − aθ = 0, a ∈ F. iii) Extension by an integral, θ0 − a = 0, a ∈ F. If this process is repeated and any finite number of elements θ1 , . . . , θk is adjoined, a Liouvillian extension Fhθ1 , . . . , θk i is obtained. The special case θ = ex , θ = log x or θ algebraic is also called an elementary extension. If a full set of solutions is not obtained in a Liouvillian extension, solutions of higher order equations have to be adjoined to the base field which are called Picard-Vessiot extensions. They may be considered as the equivalent of the splitting field of an algebraic equation. A systematic study of the possible field extensions is based on the concept of an irreducible linear ode, and the factorization of reducible equations into irreducible components. A basic prerequisite for this discussion is a theory for solving certain nonlinear equations due to Riccati that are discussed next. Solving Riccati Equations. Any n-th order linear homogeneous ode (2.1) may be transformed into a nonlinear equation of order n − 1 for a function z(x) by means of the variable change y 0 = zy. The higher order derivatives of y are then y (ν) = φν (z)y where φν (z) =

dφν−1 (z) + zφν−1 (z), ν ≥ 1 and φ0 = 1. dx

(2.10)

For ν ≤ 4 their explicit form is φ0 = 1, φ1 = z, φ2 = z 0 + z 2 , φ3 = z 00 + 3zz 0 + z 3 , φ4 = z 000 + 3z 02 + 4zz 00 + 6z 0 z 2 + z 4 . Substitution into (2.1) yields the Riccati equation associated to (2.1) Rnx (z) ≡ q0 φn + q1 φn−1 + q2 φn−2 + . . . + qn−1 φ1 + qn φ0 = 0

(2.11)

of order n − 1 for the unknown function z(x); the qk (x) for k = 1, . . . , n are the same as in (2.1), q0 = 1. For n = 2, 3 and 4 these equations are explicitly given by z 0 + z 2 + q1 z + q2 = 0, (2.12) z 00 + 3zz 0 + z 3 + q1 (z 0 + z 2 ) + q2 z + q3 = 0, 000

00

02

0 2

4

00

0

3

0

2

(2.13)

z + 4zz + 3z + 6z z + z + q1 (z + 3zz + z ) + q2 (z + z ) + q3 z + q4 = 0. (2.14)

Linear Differential Equations

17

Similar to linear equations, the existence of rational solutions of any Riccati equation is of fundamental interest. If the general solution of an equation of order n is rational it contains n constants. If this is not true, rational solutions involving fewer constants or no constants at all may exist. The latter solutions are called special rational solutions. In order to make this more precise first it has to be made clear when two solutions are considered as essentially different. To this end the following concept of equivalence of rational functions is introduced. Two rational functions p, q ∈ Q(x) are called equivalent if there exists another rational function 0 r ∈ Q(x) such that p − q = rr holds, i.e., if p and q differ only by the logarithmic derivative of another rational function. It is easily seen that this defines an equivalence relation on Q(x). Applying this concept, the following estimate of essentially different rational solutions of a Riccati equation may be obtained. Lemma 2.1 A generalized Riccati equation of order n has at most n + 1 pairwise non equivalent rational solutions. The proof of this result may be found in the article by Li and Schwarz [104]. For equations of order one and two a detailed description of all possible rational solutions is given next. Theorem 2.1 If a first order Riccati equation z 0 + z 2 + az + b = 0 with a, b ∈ Q(x) has rational solutions, one of the following cases applies: i) The general solution is rational and has the form z=

r0 +p r+C

(2.15)

¯ ¯ is a suitable algebraic extension of Q, and C is a where p, r ∈ Q(x), Q constant. ii) There is only one, or there are two inequivalent special rational solutions. Proof Let y1 , y2 be a fundamental system for the linear homogeneous equation y 00 + ay 0 + by = 0 corresponding to the given Riccati equation y0 y0 z 0 + z 2 + az + b = 0. If both y11 and y22 are not rational, a rational soluy0 y0 y0 tion does not exist. If y11 is rational but y22 is not, y11 is the only rational y0 y0 solution. If both y11 and y22 are rational but C1 y10 + C2 y20 C1 y1 + C2 y2

(2.16)

18 y0 y0 is not, y11 and y22 are two special rational solutions of the Riccati equation. y0 y0 This covers case ii). If y11 , y22 and the ratio (2.16) are rational, it may be rewritten as C1 y10 + C2 y20 Cy10 /y1 + y20 /y1 (y2 /y1 )0 y0 = = + 1 C1 y1 + C2 y2 C + y2 /y1 C + y2 /y1 y1

(2.17)

y10 y2 1 with C = C C2 . The assignments p = y1 and r = y1 yield the expressions given for case i). For first order Riccati equations there exists an extension of the preceding theorem for which there is no analogue if the order is higher than one. Corollary 2.1 The general first order Riccati equation y 0 +py 2 +qy+r = 0 may be transformed into normal form z 0 +z 2 +az+b = 0 by the variable change p0 y = zp with the result a = q − p , b = pr. The following three cases may be distinguished, depending on the number of special solutions of the normal form equation that are already known. 1 i) If a single special solution z1 is known, the general solution is z = z1 + w 0 where w is the general solution of w − (2z1 + a)w = 1. ii) If two special solutions z1 and z2 are known, the general solution is 0 R z = C r+ r + z1 with r = − exp ( (z1 − z2 )dx). iii) If three special solutions z1 , z2 and z3 are known, the general solution z2 z3 − z2 may be written as zz − − z1 : z3 − z1 = C or z3 (z1 − z2 ) zz1 − 2 − z3 z= z3 + z1 . C − zz1 − − z3 2 From these representations it follows in particular that two special rational solutions yield the general rational solution if z1 and z2 are equivalent, i.e., if the difference z1 − z2 is a logarithmic derivative. If three special rational solutions are known, the general solution is always rational. The proof is obtained by substituting the various solutions into the given equation. It is instructive to consider various examples corresponding to the alternatives of the above results. Example 2.3 The equation considered in Example 2.5 below has the two 2 . Because the difference of the special rational solutions z = x and z = x + x two special solutions may be written as a logarithmic derivative in the form 2 0 2 = (x ) , the preceding discussion shows that the general solution must be x x2

Linear Differential Equations

19

2 , the same expression for the general rational. With z1 = x and z2 = x + x 2 and z = x is chosen, its form is solution is obtained as before. If z1 = x + x 2 2 3 2 x +x+ . z= 1 x C¯ − 2 x The two constants are related by C C¯ + 1 = 0. 1
z + 1 = 0 has the two special Example 2.4 The equation z 0 + z 2 − 1 + x x 1 . The difference z − z = x is not a rational solutions z1 = 1, z2 = x + 1 2 1 x+1 logarithmic derivative. As a consequence, the general solution z=

xe−x +1 C − (x + 1)e−x

is not rational. Theorem 2.2

If a second order Riccati equation z 00 + 3zz 0 + z 3 + a(z 0 + z 2 ) + bz + c = 0

with a, b, c ∈ Q(x) has rational solutions, one of the following cases applies. i) The general solution is rational and has the form z=

C2 u0 + v 0 +p C1 + C2 u + v

(2.18)

¯ ¯ is a suitable algebraic extension of Q, and C1 Q where p, u, v ∈ Q(x), and C2 are constants. ii) There is a single rational solution containing a constant; it has the form (2.15). iii) There is a rational solution containing a single constant as in the preceding case, and in addition a single special rational solution. iv) There is only a single one, or there are two or three special rational solutions that are pairwise inequivalent. The proof for the second order equation proceeds along similar lines. Therefore it is left as Exercise 2.6. For Riccati equations of order higher than two the general structure of rational solutions has been described by Li and Schwarz [104]. Their result is given here without proof. Theorem 2.3 The rational solutions of the Riccati equation (2.11) associated to (2.1) may be described as follows. Let ri ∈ Q(x) and Pi ≡

20 {pi,1 , . . . , pi,k }, pi,j ∈ Q[x], Ci,j constants, the sets Pi linearly independent over constants for i = 1, . . . , m and j = 1, . . . , k, and define n Ci,1 p0i,1 + . . . + Ci,k p0i,k o . SPrii ≡ ri + Ci,1 pi,1 + . . . + Ci,k pi,k The rational solutions of (2.11) are the disjoint union of SPrii , i = 1, . . . , m. Pm Moreover, i=1 |Pi | is not greater than n. Due to its connection to many problems treated later on, a constructive procedure for determining the rational solutions of a Riccati equation is needed. Similar to linear equations, the first step is to find the position of its singularities. For this purpose the explicit expression φν (z) =

X k0 +2k1 +...+νkν−1 =ν

 z k0  z 0 k1  z (ν−1) kν−1 ν! ... k0 !k1 ! . . . kν−1 ! 1 2! ν!

(2.19) for the functions φν (z) in (2.11) is useful. It follows from a formal analogy to the iterated chain rule of di Bruno [22]. For the subsequent discussion several of its properties are required, e. g. the order of a pole of φν which it exhibits as a consequence of a pole of z. These properties are discussed in Exercise 2.13 and are taken for granted from now on without mentioning it. Let x0 be the position of a finite pole of order M > 1 in a possible solution 1 z(x). Due to (2.19) it generates exactly one term proportional to (x − x0 )νM in φν (z). In order to match it with some other term, the same pole must occur at least in a single coefficient qk (x) at the left hand side of equation (2.11). From this it follows that poles of order higher than one may occur only at the pole positions of the coefficients. The same is true for a pole of any order N at infinity. A bound for the order at any of these singularities is obtained by determining the growth for the various terms and looking for the largest integer where at least two terms have the same value. Due to the nonlinearity of the Riccati equation, the equations for the coefficients of the various singularities in a partial fraction expansion obey nonlinear algebraic equations that may have nonrational numbers as solutions. Therefore rational solutions of Riccati equations are searched in appropriate algebraic extensions ¯ of the rational numbers. Q These arguments do not apply for finite poles of order one. Any term in (2.19) generates a pole of order ν in this case. Nonleading terms originating from coefficient singularities may interfere with them. As a consequence, the finite first order poles are not confined to those poles occuring in the coefficients. It is a new problem to find their positions. According to Markoff [130] or Singer [169] one may proceed as follows. Let z be the desired solution of an nth order Riccati equation. Due to the fact that the residues of the additional first order poles are natural numbers, their contribution appears as a polynomial factor p in the solution of the corresponding linear equation. If the factor

Linear Differential Equations

21

originating from the higher order coefficient singularities is already known by the methods described above, a linear homogeneous equation for p may be obtained; and its possible polynomial solutions determine the additional first order poles of z. These polynomial solutions may be obtained by specializing the algorithm RationalSolutionsLode. The subsequent algorithm is designed accordingly. Algorithm 2.2 RationalSolutionsRiccati(R). Given a Riccati equation R ¯ of order n, return the rational solutions in Q(x). S1 : Higher order poles and bounds. Determine the possible positions of poles of order higher than one that may occur in a solution of R and determine an upper bound for each of them. S2 : Solve algebraic system. Set up an algebraic system for the undetermined coefficients of an ansatz within the limits found in S1, determine the coefficients from it and construct the corresponding solutions of R. If a solution does not exist, return an empty list. S3 : First order poles. For each solution obtained in S2 generate the corresponding linear equation for a function p and determine its polynomial solutions. S4 : Construct solution. From each polynomial solution obtained in S3 and the corresponding solution obtained in S2 construct a solution of the given Riccati equation. S5 : Return result. Discard those solutions obtained in S4 that are subcases of some other solution, and return a list with the remaining ones. 1
z + x2 = 0. A Example 2.5 Consider the equation z 0 + z 2 − 2x + x power xN in a possible solution generates terms of order N − 1, 2N , N + 1, N − 1 and 2, i.e., N = 1 is the largest value where at least two terms match each other. A pole at zero of the form 1M leads to singular terms of order x M + 1, 2M , M − 1, M + 1 and M − 2, therefore again M = 1 is the desired c yields the algebraic system bound. The ansatz z = ax + b + x a2 − 2a + 1 = 0, ab − b = 0, b2 + 2ac − 2c = 0, 2bc − b = 0, c2 − 2c = 0 with the two solutions a = 1, b = c = 0 and a = 1, b = 0, c = 2, i.e., two special 2 . The corresponding equations for rational solutions are z = x and z = x + x 3 p0 = 0 and p00 − 1 p0 = 0. The former equation a polynomial factor are p00 + x x does not have any polynomial solution while a fundamental system for the latter is {1, x2 }. This yields the general solution z = 22x + x. x +C Gcrd, Lclm and Factorization of Linear ODE’s. Typical questions that arise when dealing with linear differential equations are the following: Is it possible to obtain some or all elements of a fundamental system for a given linear ode as solutions of equations that are simpler than the given one? Without further specification this means that only equations of the same type,

22 i.e., linear homogeneous ode’s over the same base field, but of lower order, are admitted. More special problems are to decide whether all solutions of a given equation are also solutions of some other equation, or whether two given equations have some solutions in common. Certainly one would like to decide these questions without determining the solutions explicitly. For algebraic equations there are well-known algorithms for answering these questions. For the subsequent discussion a more algebraic language is appropriate. To this end, let Q(x)[D] with D = d be the ring of ordinary differential operdx ators in the indeterminate x with rational function coefficients. Its elements have the form A ≡ a0 Dn + a1 Dn−1 + . . . + an−1 D + an with ak ∈ Q(x). If a0 6= 0 the order of A is n. The linear ode (2.1) is obtained by applying such an operator with a0 = 1, ai = qi for i = 1, . . . , n, to a differential indeterminate y, i.e., by writing Ay = 0. Let B ≡ bm Dm + b1 Dm−1 + . . . + bm−1 D + bm be another operator. The sum A + B of two operators is defined by termwise addition in the obvious way. The product AB is defined by the prescription Dbj (x) = bj (x)D + b0j (x) from which the general expression ai (x)Di bj (x)Dj =

i   X i k=0

k

(k)

ai (x)bj Di−k+j

follows. These definitions make the set of differential operators Q(x)[D] into a ring. In general the product AB is different from BA, i.e., this ring is not commutative. Consequently, one has to distinguish between left and right factors in a product. In general, for differential operators A, B and C, C = A B implies ker(B) ⊂ ker(C), i.e., the solution space of B is a subspace of the solution space of C. Usually differential operators or linear differential equations are assumed to be primitive, i.e., the coefficient of the highest derivative is unity. For the problems mentioned above, the existence of a division algorithm is essential. It is defined in complete analogy to the corresponding divison in the ring of univariate algebraic polynomials. As a consequence, if A and B are operators of order n and m respectively with m ≤ n, there are unique operators Q and R such that A = QB + R where the order of R is strictly less than m. This guarantees that the ring Q(x)[D] is Noetherian as shown by Ore [143]. When R = 0, B is a right factor of A and Q is called the exact quotient of A and B. 1 D − 1 − 1 and B = D − 1 be given. The Example 2.6 Let A = D2 + x 4 2x 2 complete division scheme for A and B is

Linear Differential Equations

23

1D − 1 − 1
: D − 1
= D + 1 + 1 D2 + x 2 2 x
4 2x 1 2 − D − 2D 1 + 1
D − 1 + 1
1 2 x
2 x 2 1 D + 1 + 1
1 + − 1 2 x 2 x 2 0 1 1 = D + 1 + 1
D − 1
. In this particular 2 This implies D + x D − 14 − 2x 2 x 2 case the remainder vanishes, i.e., the operator A may be written as a product 1 y 0 − 1 + 1
y = 0 with the right factor B. Consequently, the equation y 00 + x 4 2x 1 y = 0. 0 has the solution exp ( x ) in common with the equation y − 2 2
3 D2 + 1 + 8 D − 2 − 10 and B = D − 2 . Example 2.7 Let A = D3 − x x x3 x x2 Proceeding as above, the representation A = QB + R, with the operators 1 D + 1 + 2 and R = 1 is obtained. This is easily verified by Q = D2 − x x2 x3 expanding the right hand side, or by going through the steps of the division scheme. The division algorithm for differential operators allows to calculate the greatest common right divisor or Gcrd(A,B) of two given operators A and B by the Euclidian algorithm, exactly like for univariate polynomials, by utilizing Gcrd(A, B) = Gcrd(B, R) if A = QB + R. Because this definition applies division on the right, it is called greatest common right divisor. In this book the greatest common left divisor which may be defined similarly will not occur. The solution space of the differential equation corresponding to the Gcrd is the intersection of the solution spaces of the differential equations corresponding to its arguments. 1 D − 1 − 1 and B = D2 + x − 1
D − x . Example 2.8 Let A = D2 + x 4 2x 2 2 1 − x + 1
D − 1
. A first division yields A = QB + R with Q = 1 and R = x 2 2 Dividing B by R yields a vanishing remainder, i.e., Gcrd(A, B) = D − 21 . This result means that the two second order linear ode’s Ay = 0 and By = 0 have 1 0 the solution exp ( x 2 ) of y − 2 y = 0 in common. Similarly the least common left multiple or Lclm(A,B) for two differential operators A and B may be defined. It is the operator of lowest order that is divided exactly by either argument from the right. The equation corresponding to the Lclm is the equation of lowest order with solution space the sum of its components. Its order is obviously at most the sum of the orders of the two arguments, and it is strictly less than that if the two arguments have a nontrivial Gcrd. Both the Gcrd and the Lclm are commutative and associative. Given any two operators, its Lclm may be determined as follows. Applying the preceding remarks, determine its order and generate an ansatz with

24 undetermined coefficients. The conditions of divisibility generate a system of linear algebraic equations, the solution of which determines the desired Lclm. Example 2.9 Let A and B as in Example 2.8. Due to the first order Gcrd, the Lclm is of order three. Dividing D3 + c1 D2 + c2 D + c3 by A and B, and equating the coefficients of the resulting remainder to zero leads to a system of linear algebraic equations for c1 , c2 and c3 . Substituting its solution into the third order ansatz finally yields   1 2 − 2x − 2 D2 Lclm(A, B) = D3 + x + x x2 − 21 x − 1   1 9 1+ 2x − 4 + 34 − x D − 14 x − 2 1 x − x−1 2

1 2

+

1 4x

+1 . x − 12 x − 1 2


From this construction it follows that exp x 2 is one solution of the corresponding third order ode, the two remaining ones are independent solutions of Ay = 0 and By = 0 respectively. The general question of reducibility of a given operator or its corresponding equation, and how to obtain its factorization into irreducible components is considered next. An operator or an equation is called irreducible if a factorization into lower order operators over the same base field does not exist. In full generality this problem has been solved at the end of the 19th century by Beke [9] and Schlesinger [160], vol. II, it requires to solve the so-called associated equations, their results are given next. Theorem 2.4

(Beke 1894, Schlesinger 1897) Let the nth order linear ode y (n) + Q1 y (n−1) + . . . + Qn y = 0

(2.20)

for y(x) be given, and let y (m) + q1 y (m−1) + . . . + qm y = 0

(2.21)


n be a possible right factor with 1 ≤ m ≤ n − 1, and N ≡ m . Further define (m) (m) w0 ≡ W (y1 , . . . , ym ) to be the Wronskian of (2.21), and wk ≡ Wk as k ; w obeys the associated equation defined by (2.3) such that qk = (−1)k w 0 w0 ¯) (N

w0

¯ −1) (N

+ r1 w0

+ . . . + rN¯ w0 = 0

(2.22)

¯ ≤ N with ri ∈ QhQ1 , . . . , Qn i for all i. The wk may be expressed of order N as ¯ −1) ¯ −2) (N (N wk = lk,N¯ −1 w0 + lk,N¯ −2 w0 + . . . + lk,0 w0 with li,j ∈ QhQ1 , . . . , Qn i for all i and j. A factor of order m of (2.20) exists w0 iff (2.22) has a hyperexponential solution, i.e., a solution such that w00 ∈ Q(x).

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25

The proof of this theorem and more details may be found in the above quoted references, see also Schwarz [162], Bronstein [21] and van Hoeij [73] where factorization algorithms are described. Example 2.10 Let n = 3 and m = 2, i.e., y 000 +Q1 y 00 +Q2 y 0 +Q3 y = 0 and y1 , y2 be a fundamental system of y 00 + q1 y 0 + q2 y = 0. If this latter equation is a factor of the former, the determinants 0 0 y y y y y y w0 = 10 20 , w1 = 001 002 and w2 = 001 002 y1 y 2 y 1 y2 y1 y2 obey w00 = w1 , w10 = w2 −Q1 w1 −Q2 w0 and w20 = −Q1 w2 +Q3 w0 . Reordering this system in a lex order with w2 > w1 > w0 yields w0000 + 2Q1 w000 + (Q01 + Q21 + Q2 )w00 + (Q02 + Q1 Q2 − Q3 )w0 = 0 w0 and w1 = w00 , w2 = w000 + Q1 w00 + Q2 w0 , i.e., explicitly q1 = − w00 and w00 w0 q2 = w00 + Q1 w00 + Q2 . In this book the general Beke-Schlesinger scheme is not considered further. Rather factorizations of second and third order equations are obtained by a direct method that leads to a more detailed description of the possible factorizations and the corresponding solutions. The subsequent lemma reduces these factorization problems to solving Riccati equations. Lemma 2.2 Determining the right irreducible factors of a linear homogeneous ode up to order three with rational function coefficients amounts to finding rational solutions of Riccati equations. i) A second order ode y 00 + Ay 0 + By = 0 has right factors of the form y 0 + ay = 0 with A, B ∈ Q(x) if and only if a ∈ Q(x) satisfies a0 − a2 + Aa − B = 0. ii) A third order ode y 000 + Ay 00 + By 0 + Cy = 0 with A, B, C ∈ Q(x) has a right factor of the form y 0 + ay = 0 if and only if a ∈ Q(x) satisfies a00 − 3aa0 + a3 + A(a0 − a2 ) + Ba − C = 0. It has a right factor of the form y 00 +by 0 +cy = 0 if and only if b, c ∈ Q(x) are solutions of a00 − 3aa0 + a3 + 2A(a0 − a2 ) + (A0 + A2 + B)a − B 0 − AB − C = 0, b = −(a0 − a2 + Aa − B). Proof

A first order factor y 0 + ay = 0 implies y 00 + (a0 − a2 )y = 0,

y 000 + (a00 − 3aa0 + a3 )y = 0.

26 Reduction w.r.t. to these constraints yields immediately the given conditions. A second order factor y 00 + ay 0 + by = 0 implies y 000 + (a0 − a2 + b)y 0 + (b0 − ab)y = 0. Reduction of the third order equation leads to the two conditions a0 − a2 + aA + b − B = 0, b0 − ab + bA − C = 0. The former one may be solved for b. If it is substituted into the latter the two conditions of the theorem are obtained. Example 2.11 Let the fourth order linear ode  1  000 L ≡ y (4) − 6x + y + 12x2 y 00 − (9x3 − 7x)y 0 + (2x4 − 6x2 )y = 0 (2.23) x be given. The corresponding third order Riccati equation has the two rational solutions z = x and z = 2x. They yield the two first order factors L1 ≡ y 0 − xy and L2 ≡ y 0 − 2xy with Lclm(L1 , L2 ) = y 00 −

3x2 + 1 0 y + 2x2 y. x

Dividing it out from L yields  2 L3 ≡ y 00 − 3xy 0 + x2 + 3 − 2 y. x This equation is irreducible. Therefore equation (2.23) has the representation L = L3 Lclm(L1 , L2 ) in terms of largest completely reducible right factors. There is an important difference between the factorization of polynomials and the factorization of differential operators, i.e., the latter is not unique as the following example shows. 2 D + 2 . Two possible Example 2.12 (Landau 1902) Consider D2 − x x2 factorizations are    1 2 1 1 + 2x  D− and D− D− . x x(1 + x) x(1 + x) More general, the factorization 

D−

 1 1 + 2ax  D− x(1 + ax) x(1 + ax)

depends on a constant parameter a.

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27

Example 2.13 Another aspect of the non-uniqueness of factorizations may be seen from the following representation of Landau’s operator as least common multiple, e. g. in the form 1 3    1 1 1 1  2 or Lclm D − − . ,D − − Lclm D − , D − 2 − 2 x x x+2 x x+1 x x−1 1 − 1
with C = C and C = C , In general, the representation Lclm D− x 1 2 C +x C1 6= C2 is valid. This means that the least common left multiple of any pair of first order operators corresponding to two different values C1 6= C2 yields Landau’s operator. The explanation of these phenomena is provided by Loewy’s results that are given next.

Loewy’s Decomposition. The factorization of differential operators into irreducible factors is not unique, as Landau’s example has shown. In order to obtain a unique decomposition of a differential operator or the corresponding linear differential equation in terms of lower order components, according to Loewy [127] or Ore [143] one has to proceed as follows. By applying a factorization algorithm, determine all irreducible right factors beginning with lowest order. The Lclm of these factors is called the largest completely reducible part, it is uniquely determined by the given operator. If its order equals n, the order of the differential operator, the operator is called completely reducible and the procedure terminates. If this is not the case, the Lclm is divided out and the same procedure is repeated with the exact quotient. After a finite number of steps the desired decomposition is obtained. By the above definition an irreducible operator is completely reducible. This proceeding is based on the following theorem by Loewy. Theorem 2.5 (Loewy 1906) Let D = d a derivative and ai ∈ F some dx differential field with derivative D. A differential operator L ≡ Dn + a1 Dn−1 + . . . + an−1 D + an of order n may be written uniquely as the product of largest completely re(d ) ducible factors Lk k of order dk over the same base field in the form (d

)

(d1 )

m−1 m) . . . L1 L = L(d m Lm−1

(dk )

with d1 + . . . + dm = n. The factors Lk

are unique.

The proof of this fundamental theorem may be found in the above quoted (k) article by Loewy. In honor of Loewy the Lj are called Loewy factors. The rightmost of them is simply called Loewy factor, it represents the largest completely reducible part. In order to apply Theorem 2.5 for generating a fundamental system of the corresponding linear ode, the further decomposition into irreducible factors is required as is described next.

28 (d )

Corollary 2.2 Any factor Lk k , k = 1, . . . , m in Theorem 2.5 may be (e ) written as the least common left multiple of irreducible operators lji i with e1 + . . . + ek = dk in the form (dk )

Lk

(e )

(e )

(e )

= Lclm(lj1 1 , lj2 2 , . . . , ljkk ).

This result provides a detailed description of the function spaces containing the solution of a reducible linear differential equation. A generalization of this result to certain systems of pde’s will be given later in this chapter. More general field extensions are studied by differential Galois theory. Good introductions are the above quoted articles by Kolchin, Singer and Bronstein, and the lecture by Magid [128]. For a fixed value of the order of a linear ode there is a finite set of possibilities differing by the number and the order of factors, some of which may contain parameters. Each alternative is called a type of Loewy decomposition. The complete answer for the most important cases n = 2 and n = 3 is listed in the following corollaries. Corollary 2.3

For n=2 the possible types of Loewy decompositions are (1) (1)

(1)

(1)

L21 : l(2) , L22 : l1 l2 , L23 : Lclm(l1 , l2 ), L24 : Lclm(l(1) (C)) where C is a constant. All types except L22 are completely reducible. The proof follows immediately from Lemma 2.2 and the classification of solutions of Riccati equations given in Theorem 2.1. A factor containing a constant corresponds to a factorization that is not unique, any special value for C generates an irreducible factor. Because the originally given operator has order 2, two different special values must be chosen in order to represent it as Lclm. Example 2.14 The examples for the various types of Loewy decompositions of second order equations are taken from Kamke’s collection, Chapter 2, they correspond to equations (2.162), (2.136), (2.201) and (2.146) respectively; as usual D = d . dx     1 y 0 + 1 − ν 2 y = D2 + 1 D + 1 − ν 2 y = 0, L21 : y 00 + x 2 x x2     x  1 1 1 1 1 2 00 0 L2 : y + x y − 4 + 2x y = D + 2 + x D − 12 y = 0,  1 0 4 L23 : y 00 + 2 + y − 2y x x  2 2x − 2 2 1  = Lclm D + − 2 , D+2+ − y = 0, 3 x x − 2x + 2 x x + 32   4 2 y = 0. L24 : y 00 − 62 y = Lclm D − 55x −x x x +C

Linear Differential Equations

29

These examples show clearly how a Loewy decomposition makes the solution space in the various functions fields completely explicit. Due to the irreducibility of all factors this representation is unique, except for type L24 decompositions which allow linear transformations over constants. The same result for equations of order three is given next. For n=3 the possible decompositions are

Corollary 2.4

(1) (1) (1)

(1)

(1)

(1)

L31 : l(3) , L32 : l(2) l(1) , L33 : l1 l2 l3 , L34 : Lclm(l1 , l2 )l3 , (1)

(1)

(1)

(1)

(1)

L35 : Lclm(l1 (C))l2 , L36 : l(1) l(2) , L37 : l1 Lclm(l2 , l3 ), (1)

(1)

(1)

(1)

(1)

L38 : l1 Lclm(l2 (C)), L39 : Lclm(l1 , l2 , l3 ), L310 : Lclm(l(2) , l(1) ), (1)

(1)

L311 : Lclm(l1 (C), l2 ), L312 : Lclm(l(1) (C1 , C2 )) where C, C1 and C2 are constants. The types L31 , and L39 through L312 are completely reducible. In the remaining cases the Loewy factor is of order less than three. Whenever a constant C appears in a factor, two different special values may be chosen such that the corresponding Lclm generates a second order operator. Similarly for the two constants in the last decomposition, three different pairs of special values may be chosen such that the Lclm generates a third order operator. Example 2.15 Most of the examples of the various types of Loewy decompositions of third order equations are again taken from Kamke’s collection, Chapter 3. The first three of them correspond to equations 3.6, 3.76 and 3.73 respectively. L31 : y 000 + 2xy 0 + y = 0,   2 + 1
D + 2 D − 2
y = 0, L32 : y 000 + 14 y 00 − 26 y = D2 + x 4 5 x x x x x 2 + 2
y 00 + 6 + 4 − 6
y 0 + 8 − 8 − 4 − 8
y L33 : y 000 − x + x x2 x x2 1 x x+1 x+1 x3


2 − 1 D − 1 D − 2 y = 0. = D− x+ x x 1 x There is no type L34 decomposition in Kamke’s collection. An example is 1 − 1 + 2
y 00 + 1 + 1 − 1
y 0 L34 : y 000 + x − x + 1 x−1 x x+1 x−1 1
Dy = 0. = Lclm D + x, D + x The next four examples are equations 3.58, 3.45, 3.37 and 3.74 respectively

30 from Kamke’s collection. 1 1 1 7 1  1 L35 : y 000 + + − 2 y 00 + + 2 y0 + 2 y 4 2x 4x x x 2x  1 1 2  1 1  D+ − = Lclm D − + − 2 y = 0, x+C x 4 2x 4x

4 y 00 + 1 + 2 y 0 + 3 y = D + 3 D2 + 1 D + 1
y = 0, L36 : y 000 + x x x x x2 1 1 0  1 1 − y + − y x−2 x x−2 x   2 1 1 = D+ + Lclm D − 1, D − y = 0, x−2 x x   1 y 00 + 1 y 0 = DLclm D − 2x L38 : y 000 − x y = 0. 2 2 x x +C

L37 : y 000 − y 00 −



The decompositions of type L39 in Kamke’s collection are more complicated. A simple example is  L39 : y 000 + x + 1 −  − 2−

2x + 1  00  2x − 1  0 2x − 2 y + x−2− 2 y x −2 x +x+1 x +x+1 2

  2x 2 2 − y = Lclm D + 1, D + x, D − y = 0. x x2 − 2 x2 + x + 1

The next example is Kamke’s equation 3.29.  3 2
1 1 L310 : y 000 + y 00 + y = Lclm D2 − 1 − D + 1 − , D + 1 + y = 0. x x x x There is no type L311 decomposition in Kamke’s collection. An example is  3 2x + 1  00 x+3 L311 : y 000 − 1 − + 2 y − 2 y0 x x +x−1 x +x−1 −

3 x

−

  3x + 4  2x 1 y = Lclm D + − , D − 1 y = 0. x x2 + C x2 + x − 1

Finally there is equation 3.71 from Kamke’s collection.       1 + 2 y 00 + 4 + 2 − 4 2 − 2 − 2 0 L312 : y 000 − x + y + 3 x 3x x2 3(x + 3) 3x x2 3(x + 3) y   2 1 x + C2 = Lclm D − 3 3x + 2C y = 0. 2 x + C1 x + C2 x + C2 In Appendix D a fairly complete list of Loewy decompositions of the linear equations listed in Chapters 2 and 3 of Kamke’s collection is given.

Linear Differential Equations

31

Example 2.16 The type L36 decomposition considered in the above example will be derived now by applying the procedure of Example 2.10. The 4 , Q = 1 + 2 and coefficients of the given third order equation are Q1 = x 2 x2 3 Q3 = x . They lead to the associated equation w0000 +

1 14  4 8 00  w0 + 1 + 2 w00 + + 3 w0 = 0 x x x x

1 . It yields q = 1 and q = 1, i.e., the second order with the solution w0 = x 1 2 x 1 y 0 + y = 0. factor y 00 + x There remains to be discussed how the solution procedure for a linear ode with a nontrivial Loewy decomposition is simplified. The general principle is described in the following corollary the proof of which is obvious. Corollary 2.5 Let a linear differential operator P of order n factor into P = QR with R of order m and Q of order n − m. Further let y1 , . . . , ym be a fundamental system for R(y) = 0, and y¯1 , . . . , y¯n−m a fundamental system for Q(y) = 0. Then a fundamental system for P (y) = 0 is given by the union of y1 , . . . , ym and special solutions of R(y) = y¯i for i = 1, . . . , n − m. Consequently, solving an inhomogeneous equation of any order with known fundamental system for the homogeneous part requires only integrations. At first this result will be applied to reducible second order equations. With D = d , y ≡ y(x) and ai ≡ ai (x), for the three nontrivial decompositions the dx following fundamental systems y1 and y2 are obtained. In decomposition L23 a1 is not equivalent to a2 as defined on page 17. L22 : (D + a2 )(D + a1 )y = 0, R R R y1 = exp (− a1 dx), y2 = y1 exp ( (a1 − a2 )dx)dx. R L23 : Lclm(D + a2 , D + a1 )y = 0, yi = exp (− ai dx), i = 1, 2, R L24 : Lclm(D + a(C))y = 0, yi = exp (− a(Ci )dx), i = 1, 2. There is an important difference between the last two cases. A fundamental system corresponding to a L23 decomposition is linearly independent over the base field, whereas this is not true for the last decomposition type L24 . It is left as Exercise 2.5 to derive these fundamental systems and their properties. An example for each decomposition type is given next. Example 2.17 The decompositions of Example 2.14 are considered again. The first equation is the well-known Bessel equation with a fundamental system Jν (x) and Iν (x) in terms of series expansions. For the type L22 equation 1 and a = − 1 , a fundamental system is y = exp x
and with a2 = 21 + x 1 1 2 2

32
R −x dx 2 e y2 = exp x x . For the type L3 decomposition, 2 4x − 4 2− 2 . 2− and a1 = 2 + x a2 = x 2x + 3 2x2 − 4x + 3 4 + 1 and y = 2 + 3 e−2x . Two independent integrations yield y1 = 32 − 3x 2 x x2 x2 4 2 , integration yields y = (x5 +C) 1 Finally in the last case a(C) = − 55x + x x +C x2 1 3 from which the fundamental system y1 = x and y2 = 2 corresponding to x C = 0 and C → ∞ follows. For differential equations of order three there are four decompositions into first order factors with no constants involved. Fundamental systems for them may be obtained as follows. R L33 : (D + a3 )(D + a2 )(D + a1 )y = 0, y1 = exp (− a1 dx), R R R R y2 = y1 exp ( (a1 − a2 )dx)dx, y3 = y1 y2 exp ( (a2 − a3 )dx)dx. R L34 : Lclm(D + a3 , D + a2 )(D + a1 )y = 0, y1 = exp (− a1 dx), R R R R y2 = y1 exp ( (a1 − a2 )dx)dx, y3 = y1 exp ( (a1 − a3 )dx)dx. R L37 : (D + a3 )Lclm(D + a2 , D + a1 )y = 0, yi = exp (− ai dx), i = 1, 2, R R R R − y2 exp ( (a2 − a3 )dx) a dx . y3 = y1 exp ( (a1 − a3 )dx) a dx 2 − a1 2 − a1 R L39 : Lclm(D + a3 , D + a2 , D + a1 )y = 0, yi = exp (− ai dx), i = 1, 2, 3. The proof of these formulas is elementary, it is based on the above Corollary 2.5 and the expression for the solution of an inhomogeneous second order linear ode given in Example 2.2. Example 2.18 The type L33 decomposition considered in Example 2.15 has 2 , a = − 1 and a = 1 − 2 . If they are substituted coefficients a1 = − x 2 3 x x x+1 in the above expressions, the fundamental system y1 = x2 , y2 = x2 log x and 2 y3 = x + x3 + x2 log (x) is obtained. Example 2.19 The type L37 decomposition considered in Example 2.15 has 2 , a = −1 and a = 1 + 1 . If they are substituted coefficients a1 = − x 2 3 x x−2 in the above expressions, the fundamental system y1 = x2 , y2 = ex and y3 =

x(x2 − 2) x2 x−2 + log + ex 4(x − 2) 4 x

Z

e−x

dx (x − 2)2

is obtained. There are various decompositions into first order factors involving one or two constants. Their fundamental systems may be obtained analogous to the preceding cases. The difference is that there may be dependencies over the

Linear Differential Equations

33

base field or an extension of it. All three elements of a fundamental system corresponding to a decomposition of type L312 are dependent over the base field. Two elements are dependent over the base field for decompositions of type L38 and L311 , and two elements of a type L35 decomposition are dependent over the simple field extension of the base field generated by its first order right factor. There are two decomposition types involving second order irreducible factors. L32 : (D2 + a3 D + a2 )(D + a1 )y = 0. Let y¯2 and y¯3 be a fundamental system of the left factor. Then Z Z
R y¯3 y¯2 dx, y3 = y1 dx. y1 = exp − a1 dx , y2 = y1 y1 y1 L36 : (D + a3 )(D2 + a2 D + a1 )y = 0. Let y1 and y2 be a fundamental system of the right factor, W = y10 y2 − y20 y1 its Wronskian. Then
y
y R R R R y3 = y1 exp − a3 dx W2 dx − y2 exp − a3 dx W1 dx. Example 2.20 position. y 000 +

Consider the following equation with the type L32 decom-

 1 x2 − 4  x2 + 1 00 4x2 − 4 0 x2 − 3 2 y + y = D + D + y + (D + x)y = 0. x x x x2 x2

The left factor corresponds to the Bessel equation for n = 2 with the fundamental system J2 (x) and Y2 (x). Applying the above formulas, the fundamental system


R R y1 = exp − 21 x2 , y2 = y1 J2 (x) exp 12 x2 dx, y3 = y1 Y2 (x) exp 12 x2 dx in terms of integrals over Bessel functions and exponentials is obtained. Symmetric Products and Symmetric Powers. The equations that are obeyed by the powerproducts of the elements of fundamental systems of given linear ode’s are important tools for understanding the field extensions generated by the solutions of certain irreducible equations. They are called the symmetric product and the symmetric power. The discussion in this subsection is based on results by Singer and Ulmer [170, 172]. Definition 2.2 (Symmetric product, symmetric power) Let M = 0 and N = 0 be linear differential equations with fundamental systems {y1 , . . . , ym } and {z1 , . . . , zn } respectively. The equation L = 0 for which the products {y1 z1 , . . . , y1 zn , . . . , ym zn } generate the solution space is called the symmetric product and is written L = M s N . If M = N , this is also called the s 2 s k symmetric power and written M . The k-th power M is defined correspondingly. Sometimes the same terminology is applied to the operators that generate the respective equations.

34 s k Singer [170] has
shown that if M has order n, the order of M is not k+n−1

s k higher than n−1 . If n = 2, the order of M is k + 1. The symmetric product is associative, i.e., there holds (L1 s L2 ) s L3 = L1 s (L2 s L3 ). Whenever a linear ode of order three or more is the symmetric power of some linear second order ode, its solution is reduced to this latter equation.

1, Let L3 ≡ y 000 − 32 y 0 + 33 y = 0 with solutions y1 = x x x y2 = x and y3 = x3 and L2 ≡ z 00 − 3 2 z = 0 with solutions z1 = √1 and x 4x √ s 2 z2 = x x. Then y1 = z12 , y2 = z1 z2 and y3 = z22 . Consequently, L3 = L 2 . Example 2.21

In order to be applicable for a solution procedure it is necessary to know a priori that a given linear ode is the symmetric power of some other equation. In Theorem 5.21 it will be seen that the Lie symmetries of a linear equation provide this information. In some applications the symmetric powers of a given linear ode have to be determined. According to the above mentioned references one may proceed as follows. In order to construct M s N , assume y and z are solutions of M and N respectively. Differentiate their product w = yz such that 0

0

0

00

00

0 0

00

w = y z + yz , w = y z + 2y z + yz , . . . , w

(k)

=

k   X k j=0

j

y (j) z (k−j)

and reduce the result w.r.t. M and N . In the resulting expressions there occur at most mn different terms y (i) z (j) with i < m, j < n. Consequently, for some k these products are linearly dependent over constants, and a linear relation for the derivatives of w results, it is the desired symmetric product equation. Its coefficients are in the base field of the original equations. 3 − 1
y = 0. Example 2.22 (Kovacic 1986) Consider the equation y 00 + x 16x2 If y1 and y2 are two solutions one obtains for the product z = y1 y2 the fol(i) (j) lowing linear system for y1 y2 , i, j = 0, 1. 2 − 3
y y = z 00 , y1 y2 = z, y10 y2 + y1 y20 = z 0 , 2y10 y20 + x 1 2 8x2

4 2 3 3 0 0 000 x − 4x2 (y1 y2 + y1 y2 ) − x2 − 4x3 y1 y2 = z . Transforming it to triangular form leads to the second symmetric power 4 − 3
z 0 + 2 − 3
z = 0. z 000 − x 4x2 x2 4x3 Computing symmetric powers may become fairly complex if the value of the power increases and the coefficients of the given equation are complicated. Therefore an explicit list of symmetric powers of a second order equation L ≡ y 00 + ry = 0 is given in Appendix B on page 379. An example is the second symmetric power z 000 + 4rz 0 + 2r0 z = 0 of y 00 + ry = 0.

Linear Differential Equations

35

Galois Theory for Second Order Equations. The structure of function fields containing the solutions of a reducible linear second order ode is obvious from its Loewy decomposition L22 , L23 or L24 . A complete answer embracing also irreducible equations is provided by the theory of Picard and Vessiot which is also called differential Galois theory. For second order equations it is explicitly described by Singer and Ulmer [172]. In the differential case, the Galois groups are subgroups of the general linear group GL2 (C) over the complex numbers C. According to Kaplansky [88], page 41, the Galois group is a unimodular group contained in SL2 (C) if the equation is in rational normal form y 00 + ry = 0. In order to obtain the possible Galois groups of such equations the algebraic subgroups of SL2 (C) must be known, they are described next. Theorem 2.6 The algebraic subgroups of SL2 (C) may be classified, up to conjugacy, as follows.    a b i) The Borel group B = , a, b ∈ C, a 6= 0 . 0 a−1    a b m ii) For m ∈ N, the groups Bm = , a, b ∈ C, a = 1 . 0 a−1    a 0 iii) The group Z = , a ∈ C, a = 6 0 . 0 a−1    a 0 m iv) For m ∈ N, the cyclic groups Zm = , a ∈ C, a = 1 . 0 a−1      a 0 0 1 v) The group D = , , a ∈ C, a = 6 0 . 0 a−1 −1 0 vi) For m ∈ N, the dihedral groups      a 0 0 1 m Dm = , , a ∈ C, a = 1 . 0 a−1 −1 0 vii) The tetrahedral group of order 24 which is generated by (θ8 = 1)  7 7      1 θ θ i 0 0 1 SL2 √ . A4 = , , 5 0 −i −1 0 2 θ θ viii) The octahedral group of order 48 which is generated by (θ8 = 1)       7 7   1 i 0 0 1 θ 0 θ θ SL2 S4 = , , , √ . 5 0 −i −1 0 0 θ7 2 θ θ

36 ix) The icosahedral group of order 120 which is generated by (η 5 = 1)    3   1 η + η4 1 η 0 SL2 A5 = − , 2 . −1 −η + η 4 0 η2 η − η3 x) The full unimodular group SL2 (C). A good reference for this result, including an introduction to the literature, is the article by Martins [131]. At first they are applied to reducible equations. Theorem 2.7 (Singer and Ulmer 1993) Let L(y) ≡ y 00 + ry = 0 be a s k reducible second order ode with r ∈ Q(x), L its k−th symmetric power for s k k ≥ 1, and sk the number of independent nontrivial rational solutions of L . The following Galois groups G(L) have to be distinguished. i) G(L) is conjugate to the Borel group B or one of its subgroups Bm if y0 and only if there is a single nontrivial solution y such that y ∈ Q(x). Furthermore, G(L) is conjugate to Bm if and only if sk = 0 for k = 1, . . . , m − 1 and sm = 1. ii) G(L) is conjugate to Z or one of its finite subgroups Zm if and only if y0 there are two independent nontrivial solutions yi such that yii ∈ Q(x) for i = 1, 2. There holds y1 y2 ∈ Q(x). Furthermore, G(L) is conjugate to Zm if and only if for k < m there holds sk = 0 for k odd, sk = 1 for k even and sm = 2 for m odd or sm = 3 for m even. The proof of this theorem may be found in the article of Singer and Ulmer [172] on page 24. The criteria for identifying Bm and Zm are based on the tabulations given in Section 6.1 and 6.2 of [131]. The theorem covers the Loewy decompositions different from L21 . The decomposition type L22 corresponds to case i), the types L23 and L24 correspond to case ii). Applying this theorem, the following result for a reducible linear ode may always be obtained: If the criteria for a fixed value of m are fulfilled, the group is either Bm or Zm . If not, the Galois group may be either of them for a value m greater than that for which the number of rational solutions of symmetric powers can be determined, or it is B or Z. In order to obtain a complete answer, a bound for the possible value of m is needed. Such a bound does not seem to be known at present. A more complete discussion of these questions may be found in the above quoted article by Martins [131]. For equations with type L24 Loewy decomposition a complete answer is always possible. Corollary 2.6 An equation with Galois group Z1 or Z2 has a type L24 Loewy decomposition. The proof of this corollary is left as Exercise 2.7.

Linear Differential Equations

37

The equation   3 1 1 y=0 L(y) ≡ y 00 + + − 4x(x − 1) 16x2 4(x − 1)2

Example 2.23

2 is discussed by Ulmer and Weil   [182], page 192. Its type  L2 Loewy decompo1 1 + 1 − 1 sition D + D− y = 0 entails case i) of 2(x − 1) 4x 2(x − 1) 4x the above theorem. A fundamental system is √ x−1 1/2 1/4 1/2 1/4 y1 = (x − 1) x , y2 = (x − 1) x log √ x+1

y0 y0 with y11 = 3x − 1 ∈ Q(x) and y22 not in the base field. Because y14 ∈ Q(x) 4x(x − 1) the Galois group is B4 . The equation L(y) ≡ y 00 + 3 2 y = 0 is discussed by Ulmer 16x and Weil [182], page 182 and page 191. Its type L23 Loewy decomposition 1 , D − 3
y = 0 yields the fundamental system y = x1/4 and Lclm D − 4x 1 4x y2 = x3/4 . There holds y1 y2 = x and y14 = x. Consequently, according to the above theorem, case ii), the Galois group is the cyclic group of order 4.         10 −1 0 i 0 −i 0 G(L) = , , , . 01 0 −1 0 −i 0 i Example 2.24

The next example is interesting because its solution requires the extension of the field of constants. Example 2.25 The equation L(y) ≡ y 00 + 7 2 y = 0 with L23 Loewy √ √16x   −3 −3 y = 0 is discussed by 2 − 2 + decomposition Lclm D − ,D − 4x 4x √

Ulmer and Weil [182], page 197. A fundamental system is y1 = x(2+i 3)/4 √ (2−i 3)/4 s k and y2 = x . For k ≤ 48, the symmetric powers L have no rational solution for k odd, and a single rational solution for k even. By the above Theorem 2.7, case ii), the conditions for Zm are not satisfied for m ≤ 48. Consequently, the Galois group is either Zm with m > 48 or Z. From the listing in Appendix D it is obvious that most second order linear ode’s given there fall under the category covered by the above theorem. The Galois groups of irreducible second order equations with type L21 decomposition are discussed next. As far as possible, the conditions for the various types of Galois groups are expressed in terms of solutions of symmetric powers. Theorem 2.8 (Singer and Ulmer 1993, Ulmer and Weil 1996) Let L(y) ≡ y 00 + ry = 0 be an irreducible second order ode with r ∈ Q(x). The following Galois groups G(L) have to be distinguished; sk is the number of independent s k nontrivial rational solutions of L = 0.

38 i) G(L) is conjugate to D if and only if there holds sk = 0 for k 6= 0 mod(4) and sk = 1 for k = 0 mod(4) for all k. It is conjugate to Dm if and only if for k < 2m the same conditions hold as in the previous case, s2m = 1 for m even and s2m+2 = 2 for m odd. 2 if and only if sk = 0 for ii) G(L) is conjugate to the tetrahedral group ASL 4 k ≤ 5, s6 = 1.

iii) G(L) is conjugate to the octahedral group S4SL2 if and only if sk = 0 for k ≤ 7, s8 = 1. 2 iv) G(L) is conjugate to the icosahedral group ASL if and only if sk = 0 5 for k ≤ 11, s6 = 12.

v) G(L) is the full group SL2 (C) if none of the preceding cases holds. These criteria allow determining a finite Galois group of an irreducible second order equation algorithmically. For the imprimitive groups D and Dm of case i) the same remarks on page 36 apply as for the groups of reducible equations covered by Theorem 2.7. In order to obtain algebraic solutions of a second order equation explicitily, the following result of Ulmer and Weil [182] is useful. Theorem 2.9 (Ulmer and Weil 1996) Let L(y) = y 00 + r(x)y = 0 be an irreducible second order ode with r ∈ k, k some differential field, and let Pm−1 y0 u ≡ y . Then the zeroes of P (u) = um + i=0 bi ui with bi ∈ k are solutions of the Riccati equation u0 + u2 + r = 0 if and only if the coefficient bm−1 is the s m negative logarithmic derivative of an exponential solution of L = 0. The coefficients bi of P (u) are determined from the system bm = 1, bi−1 =

  1 (i + 1)rbi+1 − b0i + bm−1 bi m+1−i

for 0 < i ≤ m − 1. In particular for m = 2 there holds b0 = r − 12 (b01 − b21 ). The proof may be found in the article by Ulmer and Weil [182]. The special case m = 2 is considered in Exercise 2.8. Example 2.26 The equation considered in Example 2.22 is irreducible whereas its second symmetric power is reducible. Consequently, its Galois 1 z = 0 of the second symmetric group is imprimitive. The right factor z 0 − 2x √ 1 and b = 1 − 1 are power yields the solution z = x from which b1 = 2x 0 16x2 x√ 1 u + 1 − 1 with solutions u = 1 ± 4 x . obtained, i.e., P (u) = u2 − 2x 1,2 x 4x 16x2 √ 1/4 ±2 x They yield the fundamental system y1,2 = x e .

Linear Differential Equations Example 2.27

39

Equation 2.290 from Kamke’s collection y 00 +

27x 3 y0 − y=0 27x2 + 4 27x2 + 4

2 is irreducible with rational normal form y¯00 + 41 405x2 − 264 y¯ = 0. Its second (27x + 4)2 3 symmetric power is reducible, its type L10 Loewy decomposition has a first 27x z = 0. Consequently, its Galois group is order right factor z 0 − 27x2 + 4 imprimitive. By Theorem 2.9

27x 3 135x2 − 16 u + 4 27x2 + 4 27x2 + 4 q   4 −1 3 2+ 4 is obtained. Its zeros u1,2 = 21 (x2 + 27 x ) x± 2 27 yield the solutions q ±1/3  4 1/4 4 y¯1,2 = (x2 + 27 ) of the equation in rational normal x + x2 + 27 q ±1/3  4 of the original form, and finally the solutions y1,2 = x + x2 + 27 P (u) = u2 +

equation are obtained. Example 2.28 Airy’s equation y 00 − xy = 0 is discussed by Singer and Ulmer [172]. Its 6th symmetric power y (7) − 56xy (5) − 140y (4) + 784x2 y 000 + 2352y 00 − 4(576x3 − 295)y 0 − 3456x2 y = 0 is irreducible as shown by these authors. By Theorem 2.8, case v) the Galois group of Airy’s equation is SL2 (C). At this point the discussion of linear ode’s is complete. For a second order equation a nontrivial Loewy decomposition provides the complete answer for its solutions in terms of integrations. This implies that the Galois group of the equation is a proper subgroup of the respective linear group. If the equation is irreducible there may be a finite Galois group leading to algebraic solutions. If the Galois group is SL2 (C), there may be some interesting information on the solutions in terms of transcendents that have been introduced in the literature in terms of so-called special functions like e. g. Bessel functions, Legendre functions etc. To recognize them amounts to solving an equivalence problem which is considered in Section 4.1 on page 170. If none of these alternatives applies, its solutions have to be investigated numerically or by graphical methods or, if it is interesting enough for large classes of problems, new transcendents may be introduced. In principle these remarks may be generalized to linear ode’s of any order. The collection of solved ode’s by Kamke [87] contains more than 400 linear equations of second order in Chapter C.2, and about 80 linear equation of third order in Chapter C.3. Most of them are listed in Appendix D of this book, and their Loewy decomposition is given if it is nontrivial, i.e., if it is not of type L21 or L31 .

40

2.2

Janet’s Algorithm

The theory of systems of linear homogeneous partial differential equations (pde’s) is of interest in its own right, independent of its applications for finding symmetries and invariants of differential equations. Any such system may be transformed in infinitely many ways by adding linear combinations of its members or derivatives thereof without changing its solution space. In general it is a difficult question whether there exist nontrivial solutions at all, or what the degree of arbitrariness of the general solution is. It may consist of a finite set of constants, in which case there is a finite dimensional solution space over the constants. This is true for the systems considered in this book originating from symmetry analysis of ode’s. On the other hand, the general solution may involve one or more functions depending on various numbers of arguments. The largest of these numbers is called the differential type, the number of functions depending on them is called the typical differential dimension by Kolchin [91], combined they have been baptized the gauge [62]. These questions were the starting point for Maurice Janet. He introduced a unique representation for these systems nowadays known as a Janet basis, similar to a Gr¨ obner basis representation of a system of algebraic equations. There are several advantages to using a Janet basis representation for a system of linear homogeneous pde’s. The degree of arbitrariness of its general solution, i.e., its gauge, is easily obtained from it. Due to the uniqueness of the coefficients of a Janet basis, additional invariants may be obtained providing further information on the solutions. Both Gr¨ obner bases and Janet bases have an important feature in common. Except in very simple cases it is virtually impossible to calculate any of them by pencil-and-paper, i.e., an efficient computer algebra implementation is crucial for handling concrete problems. A first example will make some of these points clear. Example 2.29 Consider the following linear homogeneous systems for two differential indeterminates z and w depending on x and y. The first system is defined by fi = 0, i = 1, . . . , 5 where f1 ≡ wy +

y 2y x zy − y1 w, f2 ≡ zxy + x wy + x zx , 2y(x2 + y) x f3 ≡ wxy − 2x y zxx − y 2 wx ,

1 w − 1 w + x z − 1 w, f4 ≡ wxy + zxy + 2y y y x y y 2y 2 f5 ≡ wyy + zxy − y1 wy + 12 w. y

(2.24)

Linear Differential Equations

41

And the second one is gi = 0, i = 1, . . . , 4 where 1z , g1 ≡ zyy + 2y y

g2 ≡ wxx + 4y 2 wy − 8y 2 zx − 8yw,

1 w − 6y 2 z , g3 ≡ wxy − 21 zxx − 2y x y 1 w + 1 w. g4 ≡ wyy − 2zxy − 2y y 2y 2

(2.25)

A priori, there does not seem to be any connection between systems (2.24) and (2.25). In particular it is not obvious whether there is any relation between the space of solutions they describe. The answer is easily obtained, however, if they are written as 2y y x e2 + e4 , f2 = ∂ e2 + x e4 + x e1 , ∂x 2y(x2 + y) ∂ f3 = ∂ e3 − 2x y ∂x e1 , ∂y 1 e − 1e + xe , f = ∂ e + ∂ e f4 = ∂ e2 + ∂ e3 + 2y 5 4 y 3 y 2 ∂x ∂y ∂x 2 ∂y 4 f1 =

and

1 e , g = ∂ e + 4y 2 e − 8y 2 e , g1 = ∂ e2 + 2y 2 2 4 1 ∂y ∂x 3 1e , g3 = ∂ e3 − 21 ∂ e1 − 6y 2 e2 − 4y 3 ∂y ∂x 1e g4 = ∂ e4 − 2 ∂ e2 + 2y 4 ∂y ∂x

(2.26)

(2.27)

where 1 w, e1 ≡ zx + 2y

e2 ≡ zy ,

e3 ≡ wx ,

e4 ≡ wy − y1 w.

(2.28)

From this representation it is clear that both systems (2.24) and (2.25) allow all solutions of the system e1 = 0, e2 = 0, e3 = 0, e4 = 0. Explicitly, they are z = C1 +C2 x and w = −2C2 y, where C1 and C2 are the integration constants, i.e., they form a two-dimensional vector space over the constants. It will be seen later on that there are no more solutions of (2.24) and (2.25). Modules over Rings of Differential Operators. From the above example it is apparent that the object of primary interest is not an individual system of pde’s, but the module that is generated from it over an appropriate ring of differential operators; they are called D-modules. This situation is similar to commutative algebra where the ideals generated by sets of polynomials are closely related to the algebraic manifolds described by the corresponding set of algebraic equations. It turns out that the language of differential algebra applied in the books by Kaplansky [88] and Kolchin [91] is the proper setting for this discussion. Good introductions to D-modules are the articles by Oaku and Shimoyama [139] and Quadrat [150]. A few basic results from these references will be given next.

42 An operator δ on a ring R is called a derivation operator if δ(a + b) = δ(a) + δ(b) and δ(ab) = δ(a)b + aδ(b) for all elements a, b ∈ R. A ring with a single derivation operator is called an ordinary differential ring. If there is a finite set ∆ of derivation operators such that δδ 0 (a) = δ 0 δ(a) for a ∈ R, δ, δ 0 ∈ ∆ the ring is called a partial differential ring. Analogous definitions apply for an ordinary differential field and a partial differential field if the ring R bis replaced by a field F. Let Θ denote the commutative semigroup generated by the elements of ∆. IfQ it is written multiplicatively, its elements can be expressed in the form θ = δ∈∆ δ e(δ) , where each P exponent e(δ) is a natural number; e(δ) = 0 defines the identity. The sum δ∈∆ e(δ) is called the order of θ. The elements of Θ are called derivative operators or simply derivatives. In this book most frequently rings of differential operators with derivation operators ∂i = ∂ and rational function coefficients will be applied, they are ∂xi denoted by D = Q(x1 , . . . , xn )[∂1 , . . . , ∂n ]. Its elements have the form X d= ri1 ,...,in (x1 , . . . , xn )∂1i1 . . . ∂nin (2.29) where ri1 ,...,in ∈ Q(x1 , . . . , xn ), almost all ri1 ,...,in equal to zero. The ring D is noncommutative with commutation rules ∂i a = a∂i + ∂a for 1 ≤ i ≤ n ∂xi and a ∈ Q(x1 , . . . , xn ). An m-dimensional left vector module Dm over D has elements (d1 , . . . , dm ) with di ∈ D for all i. The sum of two elements of Dm is defined by componentwise addition, multiplication with ring elements d by d(d1 , . . . , dm ) = (dd1 , . . . , ddm ). The relation between the submodules of Dm and systems of linear pde’s is established as follows. Let (u1 , . . . , um )T be an m-dimensional column vector of differential indeterminates. Then the product (d1 , . . . , dm )(u1 , . . . , um )T = d1 u1 + d2 u2 + . . . + dm um

(2.30)

defines a linear differential polynomial in the u’s that may be considered as the left hand side of a partial differential equation; u1 , . . . , um are called the dependent variables or functions, depending on the independent variables x1 , . . . , xn . A N × m matrix {ci,j }, i = 1, . . . , N , j = 1, . . . , m, ci,j ∈ D, defines a system of N linear homogeneous pde’s ci,1 u1 + . . . + ci,m um = 0,

i = 1, . . . , N.

(2.31)

The ith equation of (2.31) corresponds to the vector (ci,1 , ci,2 , . . . , ci,m ) ∈ Dm for i = 1, . . . , N.

(2.32)

This correspondence between the elements of Dm , the differential polynomials (2.30) and its associated pde’s(2.31) allows turning from one representation to

Linear Differential Equations

43

the other whenever it is appropriate. The module generated by these vectors determines the solutions of the system. Example 2.30 

In Example 2.29 there holds n = m = 2. The elements

  y 2y   2x 2  1 x x ∂ + ∂ , ∂y − , ∂ , ∂ , ∂ , ∂ − ∂ , xy x y y x xy y 2y(x2 + y) x x y x y2 

∂xy +

 1 1 x   2 1 1 1 ∂y − ∂x − 2 , ∂xy + ∂y , ∂y − ∂y + 2 , ∂xy 2y y y y 2y y

generate the submodule of D2 corresponding to system (2.24). The same submodule is generated by 1   1  , ∂x , (0, ∂y ), (∂x , 0), ∂y − , 0 2y y corresponding to system (2.28). The
respective equations are obtained by applying the module elements to wz with w, z differential indeterminates with nonvanishing derivatives w.r.t. x and y. Given any module M ⊂ Dm , there are two fundamental problems associated with it.  Membership in M. Establish a procedure for deciding membership in M. This problem will be solved by constructing a Janet basis for M.  Structure of M. Determine all maximal submodules N with the property M ⊂ N ⊆ Dm . Answering these questions will be the subject of the remaining part of this section. Term Orders and Rankings. Just like an ordering of monomials is a fundamental prerequisite for defining a Gr¨obner basis in commutative algebra, an ordering of partial derivatives is necessary for defining canonical forms for differential operators and differential equations. At first it will be given in the language of differential polynomials and pde’s. Definition 2.3 (Ranking) A ranking of derivatives is a total ordering such that for any derivatives δ, δ1 and δ2 , and any derivation operator θ there holds δ ≤ θδ, δ1 ≤ δ2 ⇒ δδ1 ≤ δδ2 . A term ti,j in a differential polynomial (2.31) is a differential indeterminate uα or a derivative of it, multiplied by an element of the base field. The left hand side of a linear homogeneous pde is a sum of terms. For such a pde, the ordering of partial derivatives defined above generates a unique order for its terms. A system of linear homogeneous pde’s with terms ti,j is always arranged in the form

44

t1,1 > t1,2 > . . . > t1,k1 ∧ t2,1 > t2,2 > . . . > t2,k2 ∧ .. . ∧ tN,1 > tN,2 > . . . > tN,kN such that the ordering relations are valid as indicated. Each line of this scheme corresponds to a differential polynomial or a differential equation of the system, its terms are arranged in decreasing order from left to right. In order to save space, sometimes several equations are arranged into a single line. In these cases, in any line the leading terms increase from left to right. The terms in the rectangular boxes are the leading terms containing the leading derivative, i.e., the leading term is that term preceding any other term in the respective differential polynomial or equation w.r.t. the given ordering. For any given system of pde’s there is a finite number of term orderings leading to different arrangements of terms, a trivial upper bound being the number of permutations of its terms. In any term ordering there is a lowest term which is always one of the dependent variables corresponding to e(δ) = 0. By Dickson’s lemma [33], Chapter 2, $ 4, any decreasing sequence of derivatives must terminate. In order to generate the above mentioned form of a system of pde’s effectively it is advantageous to have a predicate to decide whether or not any pair of derivatives is in correct order. To this end, a matrix of weights (matrice de cˆ otes) has been introduced by Riquier [153]. The subsequent discussion follows closely his original work and a more complete and systematic treatment of the subject by Thomas [176]. First of all a fixed enumeration (u1 , . . . , um ) is chosen for the dependent and (x1 , . . . , xn ) for the independent variables. To any derivative of a function uα a matrix I of weights is attached comprising a single column as follows. ∂uα ←→ I ≡ (i1 , i2 , . . . , in , 0, . . . , 1, . . . , 0)T . . . . ∂xinn

∂xi11 ∂xi22

(2.33)

The position of the element 1 is the (n + α)th row. The function uα itself corresponds to a matrix with i1 = i2 = . . . = in = 0. For a partial derivative (2.33), (i1 , . . . , in ) is called the derivative vector. Let J be the matrix of weights associated with some other derivative of a function uβ . Riquier’s method of placing these derivatives in a definite order is based on the following rules. Let there be given a matrix M with n + m columns corresponding to the independent and the dependent variables, and any number of rows. By definition the derivative represented by I is higher than the one represented

Linear Differential Equations

45

by J if the first nonzero element in the column M (I − J) is positive, and vice versa if it is negative. The particular form of the matrix M determines the type of ordering. Thomas [153] discussed the possible selections in full generality. He described the most general matrix leading to a given ordering and gave a canonical form for a single representative. For the applications in this book, basically three types of orderings are applied, a lexicographic ordering, a graded lexicographic ordering and a graded reverse lexicographic ordering abbreviated by lex, grlex and grevlex respectively. In the lexicographic ordering, derivatives are ordered first by functions such that ui > uj if i < j in the arrangement (u1 , . . . , um ). If the functions contained in two derivatives are identical to each other, the order of its partial derivatives is determined like in the lexicographic term order of two algebraic monomials, i.e., xi > xj if i < j in the arrangement (x1 , . . . , xn ). The power of an individual variable in the algebraic case corresponds to the order of the derivative w.r.t. this variable. The ordering according to these rules is achieved by the matrix   0 0 ... 0 m m − 1 ... 1 1 0 ... 0 0 0 ... 0   0 1 ... 0 0 0 ... 0 Mlex =     .. .. ..  . . . ...  0 0 ... 1 0

0

... 0

whereupon the lower left n × n corner is the n-dimensional unit matrix. The graded lexicographic ordering grlex is obtained if the total orders of the two derivatives are compared first. If they are different from each other, the higher one precedes the other. If not, the above lex order is applied. The corresponding matrix Mgrlex is obtained from Mlex if the first line {1 1 . . . 1 0 0 . . . 0} is included, i.e., if it has the form   1 1 ... 1 0 0 ... 0 0 0 ... 0 m m − 1 ... 1   1 0 ... 0 0 0 ... 0   Mgrlex =  0 1 . . . 0 0 . 0 ... 0    .. ..  ..  . . . ...  0 0 ... 1 0 0 ... 0 Finally, the graded reverse lexicographic ordering grevlex at first compares the total order of the derivatives like in the grlex ordering. If they are equal to each other, they are ordered by the functions. If they are identical, derivative orders w.r.t. individual variables are compared, beginning with the last variable. The first pair with a different value decides the order, the derivative with the lower value is higher than the other one. The following matrix generates this ordering.

46 

Mgrevlex

 1 ... 1 0 0 ... 0  0 ... 0 m m − 1 ... 1    0 . . . −1 0 0 ... 0   = . .. .. ..  . . . ...     0 −1 . . . 0 0 0 ... 0 −1 0 . . . 1 0 0 ... 0 1 0 0

There are numerous variations of these orderings in addition to the obvious permutations of dependent and independent variables among themselves. In any lexicographic ordering, for example, the independent variables may be compared ahead of the dependent ones, or a combination of lex and grlex orderings may be applied. The generation of orderings by weight matrices provides an easy algorithmic way for establishing it for any number of derivatives. The following algorithm Higher? establishes a reflexive ordering of partial derivatives if a matrix of weights is given. Because in a linear homogeneous system each term contains a unique derivative, it may also be applied for ordering the terms of its members. Algorithm 2.3 Higher?(d1 , d2 , M ). Given two derivatives d1 and d2 , and a matrix M of weights, true is returned if d2 does not precede d1 . S1 : Equality? If d1 = d2 return true. S2 : Generate cotes. Set up matrices I1 and I2 for d1 and d2 according to (2.33). S3 : Apply M. Generate the column M (I1 − I2 ). If the first nonvanishing element is positive, return true and f alse otherwise. Example 2.31 Consider a problem with m = n = 2, comprising two functions (w, z) depending on (y, x). For the lexicographic orderings w > z, y > x and derivatives not higher than two, (2.33) yields z   0 0   0 1

zx zy zxx zxy zyy       1 0 2 1 0 0 1 0 1 2       0 0 0 0 0 1 1 1 1 1

for the derivatives of z. The corresponding vectors for w are obtained if the last two elements 0, 1 in each column are replaced by 1, 0. Then the matrix   1100 0 0 2 1  M = 1 0 0 0 0100 generates the graded lexicographic ordering wyy > wxy > wxx > zyy > zxy > zxx > wy > wx > zy > zx > w > z.

Linear Differential Equations

47

The matrix 

 0021 M = 1 0 0 0 0100 generates the ordering wyy > wxy > wxx > wy > wx > w > zyy > zxy > zxx > zy > zx > z which is a lexicographic ordering. The orderings described so far may be uniquely translated into module orderings for elements of Dm if the transformation rules described in the preceding subsection on pages 41-43 are applied. Example 2.32 The orderings of the preceding example are reconsidered. Applying the above rules, for w > z, y > x, w = (1, 0) and z = (0, 1) the graded lexicographic ordering leads to (∂yy , 0) > (∂xy , 0) > (∂xx , 0) > (0, ∂yy ) > (0, ∂xy ) > (0, ∂xx ) > (∂y , 0) > (∂x , 0) > (0, ∂y ) > (0, ∂x ) > (1, 0) > (0, 1) for the elements of D2 of order not higher than two. Similarly the lexicographic ordering yields (∂yy , 0) > (∂xy , 0) > (∂xx , 0) > (∂y , 0) > (∂x , 0) > (1, 0) > (0, ∂yy ) > (0, ∂xy ) > (0, ∂xx ) > (0, ∂y ) > (0, ∂x ) > (0, 1). The latter order considered in this example is the P OT order which has been introduced by Adams and Loustaunau [3] for modules over polynomial rings. Their T OP order does not have an exact counterpart among the orders discussed above. Reduction and Autoreduction. The first fundamental operation of Janet’s algorithm is the reduction. Given any pair of linear differential polynomials e1 and e2 , it may occur that the leading derivative of e2 , or of a suitable derivative ∂e2 of it, equals some derivative in e1 . This coincidence may be applied to remove the respective term from e1 by an operation which is called a reduction step. To this end, multiply the appropriate derivative ∂e2 by the coefficient of its leading derivative in the term containing it in e1 , and subtract it from e1 multiplied by the leading coefficient of e2 . If e2 is monic which is usually the case, this latter operation may be skipped, otherwise the result has to be made monic. In general, reductions may occur several times. However, due to the properties of the term orderings described above and the genuine lowering of terms in any reduction step, by Dickson’s lemma the following algorithm always terminates after a finite number of iterations. After its completion no further reductions are possible. In this case e1 is called completely reduced

48 with respect to e2 . For any linear differential polynomial e the abbreviations Lder(e),

Lterm(e),

Lcoef (e),

Lf un(e)

are applied to denote the leading derivative, the leading term, the leading coefficient or the leading function of e. Coeff icient(e, t) denotes the coefficient of the term t contained in e. Algorithm 2.4 Reduce(e1 , e2 ). Given two monic linear differential polynomials e1 and e2 , e1 is returned completely reduced w.r.t. e2 . S1 : Reduction possible? Search for a term t in e1 the derivative of which may be obtained from the leading derivative of e2 by applying some appropriate ∂. If none is found, return e1 . S2 : Reduction step. Set e1 := e1 − Coeff icient(e1 , t)∂e2 and e1 := M onic e1 , then goto S1. 2 x−y 1 z and f ≡ z + 1 z. Let e1 ≡ zy − x2 zx − 2 z, f1 ≡ zx + x 2 y y y y Reduction of e1 w.r.t. f1 yields in a single step zy + y1 z, whereas reduction

Example 2.33

2 w.r.t. f2 yields − x2 zx − x2 z. y y

For some applications, a slightly more general algorithm for reduction is desirable, reducing a given equation w.r.t. to a system of pde’s instead of a single one. Algorithm 2.5 Reduce(e, S). Given a linear differential polynomial e and a system of linear monic polynomials S = {f1 , f2 , . . .}, e is returned reduced w.r.t. S. S1 : Reduction possible? Search for a polynomial fi ∈ S w.r.t. which e may be reduced. If none is found, return e. S2 : Reduction performed. Set e := Reduce(e, fi ) and goto to S1. It is obvious that its action is achieved by repeated application of the preceding algorithm Reduce. Termination follows from the finiteness of the system S and the termination of the former algorithm. If a differential polynomial has been reduced w.r.t. to a given system, it is called in normal form w.r.t. to this system. In general the result is not unique. Example 2.34 Let e1 , f1 and f2 be defined as in Example 2.33. If now e1 is reduced w.r.t. to the system {f1 , f2 }, the result is zero, independent of the order in which the reductions are performed. Example 2.35 Consider f4 in (2.24) and its reduction w.r.t. the system comprising f1 and f2 . Reduction w.r.t. to f1 is achieved in two steps. At first ∂f1 the leading derivative wxy is removed by subtraction of . After division ∂x

Linear Differential Equations

49

by the leading coefficient the result is f4a ≡ zxy +

x2 + y wy 2x y − x + 2y 2 2

+

x2 + y 2x5 + 4x3 y + x2 + 2xy 2 − y z − w. y (2x2 y − x + 2y 2 )(x2 + y) (2x2 y − x + 2y 2 )y

In the second step the term proportional to wy is removed by subtraction of a proper multiple of f1 . This leads to f4b ≡ zxy +

4x5 y + 8x3 y 2 − x3 + 2x2 y + 4xy 3 − xy − 2y 2 zy . (2x2 y − x + 2y 2 )(x2 + y)y

The result f4b may be further reduced w.r.t. to f2 . This yields f4c ≡ wy −

4x6 y + 8x4 y 2 − x4 + 2x3 y + 4x2 y 3 − x2 y − 2xy 2 zy + 2zx . (2x2 y − x + 2y 2 )(x2 + y)y 2

Now there is again a reduction possible w.r.t. f1 , which yields the answer f4d ≡ zy −

−

4(2x2 y − x + 2y 2 )(x2 + y)y 2 zx (4x y + 8x y − x3 + 2x2 y 2 + 2x2 y + 4xy 3 − 2xy + 2y 3 − 2y 2 )x 5

3 2

(2x2 y − x + 2y 2 )(x2 + y)y w = 0. (4x5 y + 8x3 y 2 − x3 + 2x2 y 2 + 2x2 y + 4xy 3 − 2xy + 2y 3 − 2y 2 )x

At this point f4d cannot be further reduced by any other equation. For any given system, in general it is possible that some elements may be reduced w.r.t. the remaining ones. Janet’s algorithm requires that at first all such reductions be performed. A system with the property that there is no reduction possible of any element w.r.t. to any other in the system is called an autoreduced system. The following algorithm accepts any ordered system of linear differential polynomials as input and returns the corresponding autoreduced system by calling the algorithm Reduce(ei , ej ) repeatedly. These calls are organized as follows. The lowest element of the given system which may be applied for any reduction is selected and all reductions w.r.t. this element are performed. The resulting system is treated in the same way until no further reduction is possible in the full system, and the result is returned. Algorithm 2.6 AutoReduce(S). Given an ordered system of linear monic differential polynomials S = {e1 , e2 , . . .}, a new system is returned, such that each element is completely reduced w.r.t. to any other member of the system. S1 : Reduction possible? Find the first element ek in S w.r.t. which any reduction may be performed in {ek+1 , ek+2 . . .}. If none is found, return the system S unchanged.

50 S2 : Perform reductions. Set S := {e1 , e2 , . . . ek } ∪ {Reduce(ej , ek )|j = k + 1, k + 2, . . .}, reorder S properly and return to S1. Termination is assured due to the finiteness of the system and the fact that any call to Reduce in step S2 generates a result with at least a single term replaced by a lower one. Example 2.36 Again system (2.24) is considered. In the first iteration of AutoReduce the lowest element f1 is applied for removing all partial derivatives of w w.r.t. y in the remaining elements with the result wy +

2y 1 w, x zy + x zx + x zy − y1 w, zxy − 2 1 2x2 y + 2y 2 2x + 2y

zxy +

4x5 y + 8x3 y 2 − x3 + 2x2 y + 4xy 3 − xy − 2y 2 zy , 4x4 y 2 − 2x3 y + 8x2 y 3 − 2xy 2 + 4y 4

zxy + (4x2 + 4y)zxx + zyy −

2x3 − 2x2 y + 2xy − 2y 2 x2 − y wx − 3 zy , xy x + xy

x2 + 2y 2x2 y + 2y 2 zxy − 2 zy . x x y + y2

The derivatives of zxy are removed by means of the second element next. wy +

zy −

−

x zy − y1 w, 2x2 y + 2y 2

8x4 y 3 − 4x3 y 2 + 16x2 y 4 − 4xy 3 + 8y 5 zx 4x y + 8x y − x4 + 2x3 y 2 + 2x3 y + 4x2 y 3 − 2x2 y + 2xy 3 − 2xy 2 6

4 2

4x4 y 2 − 2x3 y + 8x2 y 3 − 2xy 2 + 4y 4 w, 4x y + 8x y − x4 + 2x3 y 2 + 2x3 y + 4x2 y 3 − 2x2 y + 2xy 3 − 2xy 2 6

4 2

y x−y 2x2 − x − 2y zxx + 2xy wx − 5 zy − 3 zx − 3 1 w, 8x + 16x3 y + 8xy 2 2x + 2xy 4x + 4xy 2y 1 w, zxy − 2 1 zy + x zx + x 2x + 2y x3 + x2 y 2 + 2xy + y 3 4x2 y 2 + 4y 3 2x2 y + 2y 2 zyy − zy + zx + w. 3 2 2 x y + xy x x2 The next six iterations generate systems that are too large to be reproduced 1 w, and an here. After one more iteration, the lowest element becomes zx + 2y additional iteration generates the final result 1 w, z , w , w + 1 w. zx + 2y y x y y Completion and Integrability Conditions. Although autoreduction already removes a lot of arbitrariness from any sytem of differential polynomials,

Linear Differential Equations

51

there are still some shortcomings which disqualify them from being considered as canonical form. In the first place, the outcome of autoreduction is highly non-unique as the result of Example 2.36 and its comparison with the autoreduced system (2.25) shows. Furthermore, differentiating in (2.25) g2 w.r.t. y and g3 w.r.t. x, and reducing the result w.r.t. to the remaining elements of the system, the following result is obtained. ∂g2 ≡ wxxy + 2ywy − 16yzx − 10w, ∂y ∂g3 1 1 2 ≡ wxxy − 1 2 zxxx − 6y zxy − 4y zxx − 4y 2 wx − 3yzy . ∂y Either of them may be considered as a representation of the leading derivative wxxy in terms of its reductum which do not coincide. It is obvious that infinitely many instances like this may be generated by suitable differentiations. These apparent inconsistencies already hold the clue on how to proceed further, i.e., an autoreduced system must be supplemented by additional equations such that these inconsistencies disappear. In order to obtain a finite problem, according to Janet [83] certain elements which make it into a complete system have to be included. To this end, at first the full system of pde’s is partitioned into subsystems with the same function in the leading derivative. By definition, a system of linear differential polynomials of this kind is called complete if the derivative vectors of the leading terms in any subsystem form a complete set in the sense that it is defined in Appendix C for degree vectors of monomials. Because differentiation corresponds to multiplication by the respective variable, the subsequently described completion algorithm is a straightforward generalization of the completion process for systems of monomials that is described in Appendix C. Algorithm 2.7 CompleteSystem(S). Given an ordered autoreduced system of linear differential polynomials S with leading derivative vectors D, the complete system corresponding to S is returned. S1 : Separate. Represent S in the form S := S1 ∪ S2 ∪ . . . ∪ Sk such that the elements in each Si contain the same leading function with leading derivatives Di . S2 : Completion. For each i set D¯i := Complete(Di ). S3 : Generate systems. For each Si generate the elements corresponding ¯ i and assign it to S¯i . to a D S4 : Generate complete system. Collect the S¯i , assign them to ¯ S¯ := {S¯1 , S¯2 , . . . S¯k } and return Reorder(S) Termination of this process is assured by the finiteness of the completion algorithm Complete(u) for systems of monomials described in Appendix C. Consider any linear homogeneous system, choose two elements with the same leading function and solve both with respect to its leading term. By

52 suitable cross differentiation it is always possible to obtain two new polynomials such that the derivatives on the left hand sides are identical to each other. Intuitively it is expected that the same should hold true for the right hand sides if all reductions w.r.t. to the remaining elements of the system are performed. This consideration justifies the following definition. Definition 2.4 (Integrability condition). Let e1 and e2 be two elements of a system of linear differential polynomials. Let ∂ and ∂ 0 be derivatives of minimal order that make their leading derivatives identical to each other. Then the constraint Lcoef (∂ 0 e2 )∂e1 − Lcoef (∂e1 )∂ 0 e2 = 0 is called an integrability condition for the system S. The integrability conditions correspond to the S-pairs in a Gr¨obner basis calculation. A priori there are infinitely many of them in any given system of pde’s. It is one of the achievements of Janet to identify a finite number of them such that the remaining ones follow. This is the subject of the next theorem. Theorem 2.10 (Janet 1920) Let S = {e1 , e2 , . . .} be an autoreduced complete system of linear differential polynomials. In order to satisfy all integrability conditions of this system it is sufficient to satisfy ∂ m1 +...+mp ej ∂ei − m1 m =0 ∂xk ∂xi1 . . . ∂xip p

(2.34)

provided that the leading derivatives of the two expressions at the left hand side are identical to each other, xk is a nonmultiplier for ei and xi1 , . . . , xip are multipliers for ej . Proof In a complete system the leading derivatives of a function parametrize the classes of all principal derivatives. Derivatives w.r.t. the multiplier variables reproduce the derivatives of each class. Therefore in order to obtain any principal derivative in two different ways, at least a single derivation w.r.t. to a nonmultiplier variable must be involved. Let xk be a nonmultiplier variable for ei . The leading derivatives of the derived element must be contained in some other class. Let the leading derivative of ej parametrize this class. Therefore it must be possible by deriving it w.r.t. to its multiplier variables a suitable number of times to generate the same leading derivative as in ∂ei . Let this requirement be fulfilled for (2.34). Due to the uniqueness ∂xk of the representation of principal derivatives, any other element satisfying this condition has the form   ∂ n1 +...+np ej ∂ei ∂ m1 +...+mp ej − = 0. n mp 1 ∂xni11 . . . ∂xipp ∂xk ∂xm i1 . . . ∂xip The above condition (2.34) assures their vanishing.

Linear Differential Equations

53

It may be said somewhat loosely that a system of pde’s for which all integrability conditions are satisfied contains all elements explicitly that are essential for determining the general solution. This property is of such a fundamental importance that a special term is introduced for it. Definition 2.5 (Janet basis). For a given term ordering an autoreduced system of linear differential polynomials is called a Janet basis if all integrability conditions reduce to zero. An autoreduced system satisfying all integrability conditions is also called coherent. The following algorithm that accepts any system of linear differential polynomials and transforms it into a new system such that all integrability conditions are satisfied is due to Janet [83]. Algorithm 2.8 JanetBasis(S). Given a system of linear differential polynomials S = {e1 , e2 , . . .}, the Janet basis corresponding to S is returned. S1 : Autoreduction. Set S := Autoreduce(S). S2 : Completion. Set S := CompleteSystem(S). S3 : Find integrability conditions. Find all pairs of leading terms vi of ei and vj of ej such that differentiation w.r.t. a nonmultiplier xik and ∂ p1 +...+pl vj and determine the multipliers xj1 , . . . , xjl leads to ∂vi = ∂xik ∂xpj11 . . . ∂xpjll integrability conditions ∂ p1 +...+pl ej ci,j = Lcoef (ej ) · ∂ei − Lcoef (ei ) · p1 . ∂xik ∂xj1 . . . ∂xpjll S4 : Reduce integrability conditions. For all ci,j set ci,j := Reduce(ci,j , S). S5 : Termination? If all ci,j are zero return S, otherwise make the assignment S := S ∪ {ci,j |ci,j 6= 0}, reorder S properly and goto S1. Proof Steps S1, S2 and S4 are finite due to the finiteness of the algorithms called there. Applying Theorem 2.10 to all subsystems corresponding to like leading functions leads to the integrability conditions in step S3. Upon reduction in step S4, the leading derivatives of any nonvanishing integrability condition are lower than any leading derivative in the current system. Therefore this process must terminate after a finite number of iterations. The above definition of a Janet basis has the advantage of being constructive, i.e., it may be applied in straightforward manner to decide whether a given system of linear differential polynomials is a Janet basis and, if this is not the case, to transform it into one. The most important property of Janet bases is established in the next theorem. Sometimes this property of a Janet basis is used as the definition. Theorem 2.11 Let J be a Janet basis generating a module M ⊂ Dm . Then an element e ∈ Dm is contained in M if and only if it may be reduced to zero by J.

54 By means of this theorem it will be shown next that the result obtained from the algorithm JanetBasis is unique. Theorem 2.12 For a given module M and a given term ordering there exists a unique Janet basis. Proof Assume for a given module M there are two different Janet bases J = {j1 , . . . , jk } and H = {h1 , . . . , hl }. Let their elements be monic and ordered such that the leading terms increase with increasing index as explained on page 44. By Theorem 2.11 there holds Reduce(ji , H) = 0 for i = 1, . . . , k and Reduce(hi , J) = 0 for i = 1, . . . , l. In the first place this requires that the leading terms of j1 and h1 are identical, because if e. g. Lterm(j1 ) < Lterm(h1 ) holds, j1 cannot be reduced by any element of H. Consequently, the leading term of Reduce(j1 , h1 ) = j1 − h1 is lower than the common leading term of both j1 and h1 and cannot be reduced any further. By Theorem 2.11 it follows that j1 − h1 = 0, i.e., j1 = h1 . Due to this equality, both j1 and h1 cannot take part in any further reduction. Therefore the same argument may be repeated with j1 and h1 omitted and so forth until the complete bases are covered. Example 2.37 As a first example consider the system (2.25) from above. It is already autoreduced. Integrability conditions can only arise from the subsystem comprising g2 with multipliers x and y, and g3 and g4 with multiplier y each. As it is already complete, there is no action in step S2. Their multipliers lead in step S3 to the integrability conditions 1 zxxx + 8y 2 wyy + wxx − 4y 2 zxy − 32yzx − 16w = 0, zxxy − 4y 2 zyy − 8yzy = 0. y In step S4 their left-hand sides are reduced to g5 ≡ zxxx + 8y 2 wyy + y1 wxx − 4y 2 zxy − 32yzx − 16w, g6 ≡ zxxy − 4y 2 zyy − 8yzy . Because the result does not vanish, the algorithm is repeated with the enlarged system comprising equations g1 , . . . , g6 . The subsystem comprising g1 , g5 , g6 is completed in step S2 to 1z , z 2 zyy + 2y y xxx + 12y zxy − 24yzx − 12w, 1z . zxxy − 6yzy = 0, zxyy + 2y xy The first element has multiplier variables x and y, for the remaining ones y is the only multiplier variable. They yield zxxy + 12y 2 zyy + 12yzy ,

zxyy +

1 1 2 zxy − 2 wy − 2 zx . 2y y y

Linear Differential Equations

55

In step S4 they are reduced to zy , wy + 2zx . The enlarged system generated in S5 is autoreduced to zx +

1 1 w, zy , wx , wy − w 2y y

at the beginning of the third pass. Because the only two integrability conditions reduce to zero, this is the final answer.

2.3

Properties of Janet Bases

From now on the emphasis will be shifted from linear differential polynomials to the systems of lpde’s that are associated with them, and their solutions in suitable function fields. Furthermore, it is assumed frequently that the dimension of these solution spaces is finite, and that both the number of dependent and of independent variables is not greater than two. These constraints are a priori known to be true for the applications later in this book. It turns out that systems with these properties allow a complete classification; and explicit solution procedures may be designed for them. Type and Order of a Janet Basis. It turns out that the leading derivatives are of extraordinary importance for any Janet basis. They are applied for defining the type of a Janet basis. Definition 2.6 (Type of a Janet basis) Let f = {f1 , . . . , fp } be a Janet basis for a fixed term order. Its leading derivatives {Lder(f1 ), . . . , Lder(fp )} define the type of the Janet basis f . The leading derivatives divide the totality of derivatives into two classes for which a special name is introduced. Definition 2.7 (Parametric and principal derivatives. Rank). Those derivatives that may be obtained from the leading derivatives of a Janet basis by suitable differentiation are called principal, the remaining ones are called parametric. The number of parametric derivatives is called its rank. The distinctive feature of a Janet basis as compared to an arbitrary system of pde’s may be expressed in terms of its parametric derivatives as follows: A Janet basis does not generate any relation between the parametric derivatives by differentiation and elimination, i.e., the parametric derivatives are independent of each other. Another quantity of importance for any system of pde’s is the degree of arbitrariness of its general solution. As a matter of fact, this question was the starting point for Janet’s work. In general it may consist of any number of functions with varying numbers of arguments. A special case is represented by

56 those systems the general solution of which contains only a finite number of constants. Due to the linearity, their general solution is a linear combination of a fundamental system, i.e., it generates a finite dimensional vector space. A straightforward extension of this property which is true for linear ode’s leads to the following definition. Definition 2.8 (Order). If the general solution of a linear homogeneous system of pde’s contains a finite number of constants, this number is called the order of the system. It is identical to the dimension of its solution space. The importance of a Janet basis originates from the fact that the dimension of the solution space is obvious from it due to the following result. Its proof may be found in the book by Kolchin [91], Chapter IV, Section 5. Theorem 2.13 (Kolchin 1973) If the rank of a Janet basis is finite, then it is identical to its order. Example 2.38 The type of the Janet basis generated in Example 2.37 is determined by {zx , zy , wx , wy }. The only parametric derivatives are z and w; therefore it allows a two-dimensional solution space as obtained previously. This information is available without determining the solution explicitly. Theorem 2.14 If the dimension of the solution space of a system of linear homogeneous pde’s is finite, any Janet basis in lexicographic term ordering has a subsystem containing only the lowest dependent variable. This subsystem also has a finite-dimensional solution space. Its lowest equation is a linear ode in the lowest independent variable. Proof Let the dependent variables be ordered as z1 > z2 . . . > zm , the independent variables as x1 > x2 > . . . > xn . The finite dimension of the solution space requires that the Janet basis contains equations with leading ki terms ∂ zkm for i = 1, . . . , n and ki ≥ 0. This assures the finite-dimensional ∂xi i solution space for the lowest dependent variable. The applied term ordering guarantees that in these equations there does not occur a term containing a function zi with i < m. Due to the ordering of the independent variables, the kn equation with leading term ∂ kznm cannot contain a derivative w.r.t. to xk ∂xn and k < n, i.e., it is an ode w.r.t. to xn . Example 2.39 Consider the Janet basis n o 1 1 1 z1,x + z2 − z1 , z1,y , z2,x , z2,y − z2 y−2 x y−2

(2.35)

in grlex order with z2 > z1 and x > y. There are four possible lex term orderings with the following result. n y−2 o 1 1 z1,xx − z1,x + 2 z1 , z1,y , z2 + (y − 2)z1,x − z1 for z2 > z1 , y > x. x x x

Linear Differential Equations

57

1 1 1 z2 , z1,x − z1 + z2 , z1,y for z1 > z2 , y > x. y−2 x y−2 n 1 1 y−2 o z1 for z2 > z1 , x > y z1,y , z1,xx − z1,x + 2 z1 , z2 + (y − 2)z1,x − x x x and finally n o 1 1 1 z2,y − z2 , z2,x , z1,y , z1,x − z1 + z2 for z1 > z2 , x > y. y−2 x y−2 o

n z2,x , z2,y −

In each of these Janet bases the two lowest equations generate a Janet basis for a single function. If y > x the lowest equation is a linear ode in x over the base field Q(y) and vice versa.

(1,2)

(1,2)

(1,2)

J1 J2,1 J2,2 zx z zy zx , z zx z zy z zxx zx , z zyy zy , z (1,2)

(1,2)

(1,2)

J3,1 J3,2 J3,3 zy zx , z zxx zy , zx , z zx z zxxx zxx , zx , z zxy zy , zx , z zyyy zyy , zy , z zyy zy , zx , z

TABLE 2.1: The types of Janet bases for m = 1, n = 2, i.e., for a single function z(x, y) of order up to 3. The term ordering is grlex with y > x. Any type is uniquely characterized by its leading derivatives in the square boxes. Classification for m=1, n=2. The simplest generalization of a linear homogeneous ode is a system of lpde’s of finite order for a single function. For any fixed value of the order and a fixed term ordering there is a finite number of possible types of Janet bases. They are obtained by the following considerations. In the grlex term ordering with y > x the derivatives of order not higher than three are zyyy > zxyy > zxxy > zxxx > zyy > zxy > zxx > zy > zx > z. According to the definition of a parametric derivative, for an r-dimensional solution space, principal derivatives must be selected from this arrangement such that none of it may be obtained by differentiation from any of the remaining ones and that they allow r parametric derivatives. For r = 1, 2 or 3 the possible selections are as follows. r=1:

{zx , zy },

r=2:

{zy , zxx }, {zx , zyy },

r=3:

{zy , zxxx }, {zxx , zyx , zyy }, {zx , zyyy }.

58 In Table 2.1 the structure of the corresponding Janet bases is indicated; (m,n) and a unique notation Jr,k for any of them is introduced. Here m and n denote the number of dependent and independent variables respectively, r denotes the order, and k is a consecutive enumeration within this set. The symbol J (m,n) without the lower indices denotes collectively all Janet basis types corresponding to the given values of m and n. Each line corresponds to an element of the Janet basis, its leading derivative is enclosed in a box, the parametric derivatives that may occur are listed to the right in decreasing order. In a concrete Janet basis they are multiplied by a coefficient from the base field. The possible arrangements of parametric and leading derivatives may also be visualized geometrically. To this end, for any dependent variable its partial derivatives are assigned to a point of a rectangular lattice. Parametric derivatives are indicated by open dots, principal derivatives by heavy dots. To the six types of Janet bases defined in Table 2.1 there correspond the following graphs. y y y 26 26 2 s6 (1,2)

J1

(1,2)

: 1s c

s

c

-

2 x

1

y

c

J2,2 : 1 c c

s

2 x

1

y 6

c

c

1

c

2

s

3 x

-

2 x

3 s6

2s (1,2) J3,2 : 1 c

1s

s

1

y 6

2 (1,2) J3,1 :

(1,2)

J2,1 : 1 s

c

2c (1,2) J3,3 :

s c

1

s

2

-

x

1c

c

s

1

2

x

The classification of Janet bases given above provides a complete survey of all possible Janet bases for m = 1, n = 2 if the dimension of the solution space is not higher than three. If it is a priori known for a particular system that the dimension of its solution space is at most three, it is guaranteed that the resulting Janet basis type is included in this listing. Next to the classification problem there occurs the following question: What are the most general coefficients for any given type such that the Janet basis property is assured? This is a new phenomenon that does not have a counterpart for linear ode’s of the form (2.1). Its coefficients qi may be chosen arbitrarily from the base field, yet the resulting equation has an n−dimensional solution space, only the function field in which a fundamental system is contained is determined by this choice. This is not true for Janet bases for systems of lpde’s. The constraints for the coefficients guaranteeing the Janet basis property are extremely important, particularly if they are combined with

Linear Differential Equations

59

additional restrictions expressing further properties of the solutions as they will occur in later applications. Only if the former constraints are completely known, the latter ones may be identified as new features originating from the special problem. The coefficient constraints for a given Janet basis type are essentially obtained in steps S2, S3 and S4 of the algorithm JanetBasis given above. Because the type of the Janet basis is fixed now, upon completion of step S3 the nonvanishing integrability conditions are not added to the system. Rather in order to preserve the type of the Janet basis, the coefficients of the various parametric derivatives are supposed to vanish identically. The conditions obtained in this way represent a system of pde’s for the coefficient functions. If it is satisfied, the full system is guaranteed to be the most general Janet basis of the specified type. This is achieved by insertion of the additional step S5a as follows. S5a : Separate Integrability Conditions. Assign S C := ci,j Coeff icients(ci,j ) and return C. In general the integrability conditions obtained in step S5a are a system of nonlinear pde’s that may comprise redundant equations. However, the following important property makes it amenable for further simplification. Lemma 2.3 The integrability conditions obtained in step S5a are quasilinear, i.e., they are linear in the highest derivatives. Proof In step S2 and the first part of S3 only differentiations of the given linear homogeneous equations are performed, i.e., the resulting system is such that the reductum of any equation is linear and homogeneous in the coefficients. The same is true for the integrability conditions ci,j obtained in the second part of S3 because the leading terms cancel each other. For any given ci,j two kinds of reductions in step S4 may occur. In the first place there are those applying an equation ek with an index different from i and j. They can never generate a quadratic term because there is no intersection between the coefficient variables of any equation involved. On the other hand, any reduction w.r.t. equation ei or ej involves only derivatives that are lower than those involved in forming the integrability condition. As a consequence any derivative of maximal order can only occur linearly. Applying the modified version of algorithm JanetBasis to the entries of Table 2.1 yields the following answer. Theorem 2.15 For m = 1, n = 2 and order r ≤ 3 the coherence conditions or integrability conditions (IC’s for short) are explicitly given by the following constraints. (1,2)

J1

: {zx + az, zy + bz}. IC : ay − bx = 0.

60 (1,2)

J2,1

{zy + a1 zx + a2 z, zxx + b1 zx + b2 z}.

:

a1,xx − b1,x a1 − a1,x b1 + 2a2,x − b1,y = 0, a2,xx + a2,x b1 − 2a1,x b2 − b2,x a1 − b2,y = 0.

IC 0 s : (1,2)

J2,2

{zx + a1 z, zyy + b1 zy + b2 z}.

:

0

IC s : b1,x − 2a1,y = 0, a1,yy + a1,y b1 − b2,x = 0. (1,2)

J3,1

{zy + a1 zx + a2 z, zxxx + b1 zxx + b2 zx + b3 z}.

:

a1,xx − 31 b1,y − 13 b1,x a1 + a2,x − 31 a1,x b1 = 0, a2,xxx + a2,xx b1 − b3,y − b3,x a1 + a2,x b2 − 3a1,x b3 = 0,

IC 0 s :

b1,xy + b1,xx a1 + 6a2,xx − 3b2,y − 3b2,x a1 + 34 b1,y b1 +2b1,x a1,x + 2a2,x b1 − 6a1,x b2 + 34 b1 (a1 b1 )x = 0. (1,2)

J3,2

:

{zxx + a1 zy + a2 zx + a3 z, zxy + b1 zy + b2 zx + b3 z, zyy + c1 zy + c2 zx + c3 z}. a1,y − b1,x + a1 b2 − a1 c1 − a2 b1 + a3 + b21 = 0, a2,y − b2,x − a1 c2 + b1 b2 − b3 = 0,

IC 0 s :

a3,y − b3,x − a1 c3 − a2 b3 + a3 b2 + b1 b3 = 0, b1,y − c1,x + a1 c2 − b1 b2 + b3 = 0, b2,y − c2,x + a2 c2 − b1 c2 − b22 + b2 c1 − c3 = 0, b3,y − c3,x + a3 c2 − b1 c3 − b2 b3 + b3 c1 = 0.

(1,2)

J3,3

IC 0 s :

Proof

{zx + a1 z, zyyy + b1 zyy + b2 zy + b3 z}.

:

b1,x − 3a1,y = 0, a1,yy − 31 b2,x + 32 a1,y b1 = 0, b2,xy − 3b3,x + 31 b2,x b1 − 2b1,y a1,y + 3a1,y b2 − 23 a1,y b21 = 0. (1,2)

For the J1

Janet basis the only IC follows by subtracting

zxy + (ay − ab)z and zxy + (bx − ab) from each other. These expressions are obtained by deriving the two elements of the Janet basis w.r.t. y and x respectively and reducing the result. The condition obtained assures the path independence of integrals involving the gradient vector field of z. (1,2) As a second example, the IC’s for the type J2,1 Janet basis will be obtained. To this end, the first element has to be derived twice w.r.t. to the

Linear Differential Equations

61

multiplier x and all possible reductions have to be performed. The result is zxy + (a1,x + a2 − a1 b1 )zx + (a2,x − a1 b2 )z, zxxy + (a1,xx + 2a2,x − 2a1,x b1 − a1 b1,x − a1 b2 − a2 b1 + a1 b21 )zx +(a2,xx − 2a1,x b2 − a1 b2,x − a2 b2 + a1 b1 b2 )z. The second element has to be derived w.r.t. to its nonmultiplier y. After all reductions are performed, the expression zxxy + (b1,y − a1,x b1 − a2 b1 − a1 b2 + a1 b21 )zx + (b2,y − a2,x b1 − a2 b2 + a1 b1 b2 )z is obtained. If the two elements with the leading derivative zxxy are subtracted from each other, the resulting expression must vanish identically. Because the parametric derivatives zx and z are independent, its coefficients must vanish. After some simplification this yields the given IC’s. The calculations for the remaining cases are similar. Example 2.40 n zxx −

Consider the set of differential polynomials o 1 1 1 y zy + zx , zxy + zy , zyy + zy . x(x + y) x x+y x+y (1,2)

Its leading terms suggest that it might be a type J3,2 nonvanishing coefficients a1 = −

Janet basis. The

y 1 1 , a2 = and b1 = c1 = x(x + y) x x+y

satisfy the conditions given in the above theorem, i.e., the Janet basis property is assured without running the Janet basis algorithm. As is easily shown, a basis for its three-dimensional solution space is {1, log x, log (x + y)}. In general, any change in the coefficients destroys the Janet basis property, e.g. if the sign of the last term in the first equation is reversed. Running (1,2) this new system through the algorithm JanetBasis yields the type J2,1 1 z }. A basis for its two-dimensional solutions space Janet basis {zy , zxx − x x  2 is 1, x . Classification for m=n=2. The second case to be discussed in full detail involves two dependent variables, they are denoted by z1 and z2 . In grlex term ordering with z2 > z1 and y > x, the derivatives of order not higher than two are . . . z2,yy > z2,yx > z2,xx > z1,yy > z1,yx > z1,xx > z2,y > z2,x > z1,y > z1,x > z2 > z1 .

62 (2,2)

J1,1 z1 z2,x , z2 z2,y , z2 (2,2)

J2,1 z1 z2,y , z2,x , z2 z2,xx , z2,x , z2

(2,2)

J2,2 z1 z2,x , z2 z2,yy , z2,y , z2

(2,2)

J1,2 z2 , z 1 z1,x , z1 z1,y , z1

(2,2)

J2,3 z1,x z1,y z2,x z2,y

(2,2)

(2,2)

J2,4 J2,5 , z 2 , z 1 z2 , z 1 z2 , z 1 , z2 , z1 z1,y , z1,x , z1 z1,x , z1 , , z2 , z1 z1,xx , z1,x , z1 , z1,yy , z1,y , z1 , z2 , z1

TABLE 2.2:

(2,2)

(2,2)

The Janet bases of types J1,k and J2,k for linear homogeneous systems in z1 (x, y) and z2 (x, y). The term ordering is grlex with z2 > z1 and y > x. Any type is uniquely characterized by its leading derivatives in the square boxes.

For an r-dimensional solution space, principal derivatives must be selected from this arrangement such that none of it may be obtained by differentiation from any of the remaining ones and such that there are r parametric derivatives. For r = 1, 2 or 3 the possible selections are as follows. r=1:

{z2 }, {z1 }

r=2:

{z2,x , z2 }, {z2,y , z2 }, {z2 , z1 }, {z1,y , z1 }, {z1,x , z1 }

r=3:

{z2,xx , z2,x , z2 }, {z2,y , z2,x , z2 }, {z2,yy , z2,y , z2 }, {z2,x , z2 , z1 }, {z2,y , z2 , z1 }, {z1,x , z2 , z1 }, {z1,y , z2 , z1 }, {z1,yy , z1,y , z1 }, {z1,y , z1,x , z1 }, {z1,xx , z1,x , z1 }.

In Tables 2.2 and 2.3 the structure of the corresponding Janet bases is indicated and a unique notation for any of them is introduced. Each line corresponds to an equation of the Janet basis, its leading derivative is enclosed in a box, the parametric derivatives that may occur in any equation are listed to the right of it in decreasing order. In a concrete Janet basis they are multiplied by a coefficient from the base field. Similar schemes may be set up for higher dimensional solution spaces or for values of m and n different from 2. The number of alternatives increases quickly if these values increase. In the graphical representation there correspond to each Janet basis as many drawings as there are dependent variables. For the five possible types (2,2) of Janet bases J2,k these graphs are as follows.

Linear Differential Equations (2,2)

J3,1 z1 z2,y , z2,x , z2 z2,xxx , z2,xx , z2,x , z2 (2,2)

J3,5 z1,x , z2 , z1 z1,y , z2 , z1 z2,x , z2 , z1 z2,yy , z2,y , z2 , z1

(2,2)

J3,2 z1 z2,xx , z2,x , z2 z2,yx , z2,y , z2,x , z2 z2,yy , z2,y , z2,x , z2 (2,2)

J3,6 z1,y , z1,x , z2 , z1 z2,x , z1,x , z2 , z1 z2,y , z1,x , z2 , z1 , z1,xx , z1,x , z2 , z1 (2,2)

J3,9 z2 , z 1 z1,xx , z1,y , z1,x , z1 z1,yx , z1,y , z1,x , z1 z1,yy , z1,y , z1,x , z1

63

(2,2)

J3,3 z1 z2,x , z2 z2,yyy , z2,yy , z2,y , z2

(2,2)

J3,7 z1,x , z2 , z1 z2,x , z1,y , z2 , z1 z2,y , z1,y , z2 , z1 z1,yy , z1,y , z2 , z1

(2,2)

J3,4 z1,x , z2 , z1 z1,y , z2 , z1 z2,y , z2,x , z2 , z1 , z2,xx , z2,x , z2 , z1

(2,2)

J3,8 z2 , z 1 z1,x , z1 z1,yyy , z1,yy , z1,y , z1

(2,2)

J3,10 z2 , z 1 z1,y , z1,x , z1 z1,xxx , z1,xx , z1,x , z1

(2,2)

TABLE 2.3:

The Janet bases of types J3,k , k = 1, . . . , 10 for linear homogeneous systems in z1 (x, y) and z2 (x, y). The term ordering is grlex with z2 > z1 and y > x. Any type is uniquely characterized by its leading derivatives in the square boxes. y

z1

26 (2,2)

y

z2

26 1s

J2,1 : 1 s

1

y

z1

26

c

-

2 x

(2,2)

z2,x > z2 c

1

y

s

2 x

z2

2 s6 1c

J2,2 : 1 s

1

y

z1

26 (2,2)

J2,3 : 1 s c

c

-

2 x

z2,y > z2 s

1

y

-

2 x

z2

26 1s

s

1

-

2 x

c

z2 > z 1 s

1

-

2 x

64 y

z1

26 (2,2)

J2,4 : 1 s c

z2 z1,x > z1

1

c

1

y

s

s

2 x

z1

2 s6

(2,2)

J2,5 : 1 c c

y 26

1

-

2 x

y

z2

26

z1,y > z1

1

s

1

-

2 x

s

1

-

2 x

From this representation the distinctive properties of parametric and nonparametric derivatives are obvious. The analogue of Theorem 2.15 for Janet bases of type J (2,2) of order not greater than three is given next. Due to the size of the result it is split into two cases, at first order one and two Janet bases are considered, and the third order case thereafter. Theorem 2.16 For m = n = 2 and order 1 or 2, the coherence conditions or integrability conditions (IC’s for short) are explicitly given by the following constraints. (2,2)

J1,1 (2,2)

J1,2

: {z1 , z2,x + az2 , z2,y + bz2 }. IC : ay − bx = 0.

: {z2 + az1 , z1,x + bz1 , z1,y + cz1 }. IC : by − cx = 0.

(2,2)

J2,1

: {z1 , z2,y + a1 z2,x + a2 z2 , z2,xx + b1 z2,x + b2 z2 }.

IC 0 s : (2,2)

J2,2

a1,xx − b1,y − b1,x a1 + 2a2,x − a1,x b1 = 0, a2,xx − b2,y − b2,x a1 + a2,x b1 − 2a1,x b2 = 0. : {z1 , z2,x + a1 z2 , z2,yy + b1 z2,y + b2 z2 }.

IC 0 s : b1,x − 2a1,y = 0, a1,yy − b2,x + a1,y b1 = 0. (2,2)

J2,3

:

{z1,x + a1 z2 + a2 z1 , z1,y + b1 z2 + b2 z1 , z2,x + c1 z2 + c2 z1 , z2,y + d1 z2 + d2 z1 }. b1,x − a1,y + a1 d1 − b1 c1 − a1 b2 + a2 b1 = 0,

IC 0 s :

b2,x − a2,y + a1 d2 − b1 c2 = 0, d1,x − c1,y − a1 d2 + b1 c2 = 0, d2,x − c2,y + c1 d2 − a2 d2 − c2 d1 + b2 c2 = 0.

(2,2) J2,4

: {z2 + a1 z1 , z1,y + b1 z1,x + b2 z1 , z1,xx + c1 z1,x + c2 z1 }. b1,xx − c1,y − c1,x b1 + 2b2,x − b1,x c1 = 0, IC 0 s : b2,xx − c2,y − c2,x b1 + b2,x c1 − 2b1,x c2 = 0.

Linear Differential Equations

65

(2,2)

J2,5 : {z2 + a1 z1 , z1,x + b1 z1 , z1,yy + c1 z1,y + c2 z1 }. IC 0 s : c1,x − 2b1,y = 0, b1,yy − c2,x + b1,y c1 = 0. Theorem 2.17 For m = n = 2 and order 3 the coherence conditions or integrability conditions (IC’s for short) are explicitly given by the following constraints. (2,2)

J3,1

{z1 , z2,y + a1 z2,x + a2 z2 , z2,xxx + b1 z2,xx + b2 z2,x + b3 z2 }.

:

a1,xx − 31 b1,y − 31 b1,x a1 + a2,x − 13 a1,x b1 = 0, b1,xy + b1,xx a1 + 6a2,xx − 3b2,y − 3b2,x a1 + 43 b1,y b1 + 2b1,x a1,x

IC 0 s :

+ 34 b1,x a1 b1 + 2a2,x b1 − 6a1,x b2 + 43 a1,x b21 = 0, a2,xxx + a2,xx b1 − b3,y − b3,x a1 + a2,x b2 − 3a1,x b3 = 0.

(2,2)

J3,2

:

{z1 , z2,xx + a1 z2,y + a2 z2,x + a3 z2 , z2,xy + b1 z2,y + b2 z2,x + b3 z2 , z2,yy + c1 z2,y + c2 z2,x + c3 z2 }. b1,x − a1,y + a1 c1 − a1 b2 − b21 + a2 b1 − a3 = 0, b2,x − a2,y + a1 c2 + b3 − b1 b2 = 0, b3,x − a3,y + a1 c3 − b1 b3 + a2 b3 − a3 b2 = 0,

IC 0 s :

c1,x − b1,y − a1 c2 − b3 + b1 b2 = 0, c2,x − b2,y + c3 + b1 c2 − a2 c2 − b2 c1 + b22 = 0, c3,x − b3,y + b1 c3 − a3 c2 − b3 c1 + b2 b3 = 0.

(2,2)

J3,3

:

IC 0 s :

(2,2)

J3,4

:

{z1 , z2,x + a1 z2 , z2,yyy + b1 z2,yy + b2 z2,y + b3 z2 }. b1,x − 3a1,y = 0, a1,yy − 31 b2,x + 23 a1,y b1 = 0, b2,xy − 3b3,x + 31 b2,x b1 − 2a1,y b1,y + 3a1,y b2 − 23 a1,y b21 = 0. {z1,x + a1 z2 + a2 z1 , z1,y + b1 z2 + b2 z1 , z2,y + c1 z2,x + c2 z2 + c3 z1 , z2,xx + d1 z1,x + d2 z2 + d3 z1 }.

a1 c1 + b1 = 0, b2,x − a2,y + a1 c3 = 0, b1,x − a1,y + a1 c2 − a1 b2 + a2 b1 = 0, c1,xx − d1,y − d1,x c1 + 2c2,x − c1,x d1 a1 c3 = 0, IC 0 s : c2,xx − d2,y − d2,x c1 − 2c3,x a1 + c2,x d1 − 2c1,x d2 −a1,x c3 − a1 c3 d1 + a1 a2 c3 = 0, c3,xx − d3,y − d3,x c1 + c3,x d1 − 2c3,x a2 − 2c1,x d3 − a2,x c3 −c2 d3 + a2 c1 d3 + b2 d3 + c3 d2 − a2 c3 d1 + a22 c3 = 0.

66 (2,2)

J3,5

{z1,x + a1 z2 + a2 z1 , z1,y + b1 z2 + b2 z1 , z2,x + c1 z2 + c2 z1 , z2,yy + d1 z2,y + d2 z2 + d3 z1 }.

:

a1 = 0, b1,x − b1 c1 + a2 b1 = 0, b2,x − a2,y − b1 c2 = 0, d1,x − 2c1,y + b1 c2 = 0, 0 IC s : c1,yy − d2,x − 2c2,y b1 + c1,y d1 − b1,y c2 − b1 c2 d1 + b1 b2 c2 = 0, c2,yy − d3,x + c2,y d1 − 2c2,y b2 − b2,y c2 −c1 d3 + a2 d3 + c2 d2 − b2 c2 d1 + c2 b22 = 0. {z1,y + a1 z1,x + a2 z2 + a3 z1 , z2,x + b1 z1,x + b2 z2 + b3 z1 , z2,y + c1 z1,x + c2 z2 + c3 z1 , z1,xx + d1 z1,x + d2 z2 + d3 z1 }. c1,x − b1,y + a1,x b1 − c1 d1 − a1 b1 d1 + c3 −b1 c2 + b2 c1 + a1 d3 − b21 a2 + a3 b1 = 0, c2,x − b2,y + a2,x b1 − c1 d2 − a1 b1 d2 + a2 d3 − a2 b1 b2 = 0, c3,x − d3,y + a3,x b1 − c1 d3 − a2 b1 d3 +b2 c3 − c2 d3 − a1 b2 d3 + a3 d3 = 0, 0 IC s : a1,xx − d1,y − d1,x a1 − b1,x a2 + 2a3,x − 2a2,x b1 −a1,x d1 + c1 d1 + a1 b1 d2 − a2 d3 + a2 b1 b2 = 0, a2,xx − d2,y − d2,x a1 − b2,x a2 + a2,x d1 − 2a2,x b2 − 2a1,x d2 +a2 d3 + c2 d2 + a1 b2 d2 + a2 b1 d2 − a3 d2 − a2 b2 d1 + b21 a2 = 0, a3,xx − d3,y − d3,x a1 − d3,x a2 + a3,x d1 − 2a2,x d3 −2a1,x d3 + a2 b1 d3 + c3 d2 + a1 d2 d3 − a2 d1 d3 + a2 b2 d3 = 0. (2,2)

J3,6

:

(2,2)

J3,7

:

{z1,x + a1 z2 + a2 z1 , z2,x + b1 z1,y + b2 z2 + b3 z1 , z2,y + c1 z1,y + c2 z2 + c3 z1 , z1,yy + d1 z1,y + d2 z2 + d3 z1 }.

c1,x − b1,y + b1 d1 − b1 c2 + c21 a1 − a2 c1 + b2 c1 − d3 = 0, c2,x − a1,y c1 + b1 d2 − a1 c3 + a1 c1 c2 − b2,y = 0, c3,x − a2,y c1 + b1 d3 + a1 c1 c3 − a2 c3 + b2 c3 − c2 d3 = 0, d1,x + c1,y a1 − 2a2,y + 2a1,y c1 − b1 d2 + a1 c3 − a1 c1 c2 = 0, IC 0 s : a1,yy − d2,x − c2,y a1 + a1,y d1 − 2a1,y c2 +a1 d3 + a1 c1 d2 − a2 d2 + b2 d2 − a1 c2 d1 + c22 a1 = 0, a2,yy − d3,x − c3,y a1 + a2,y d1 − 2a1,y c3 +a1 c1 d3 + d2 d3 − a1 c3 d1 + a1 c2 c3 = 0. (2,2)

J3,8

: {z2 + a1 z1 , z1,x + b1 z1 , z1,yyy + c1 z1,yy + c2 z1,y + c3 z1 }.

IC 0 s :

c1,x − 3b1,y = 0, b1,yy − 31 c2,x + 23 b1,y c1 = 0, c2,xy − 3c3,x + 13 c3,x c1 − 2c1,y b1,y + 3b1,y c2 − 23 b1,y c21 = 0.

Linear Differential Equations

67

{z2 + a1 z1 , z1,xx + b1 z1,y + b2 z1,x + b3 z1 , z1,xy + c1 z1,y + c2 z1,x + c3 z1 , z1,yy + d1 z1,y + d2 z1,x + d3 z1 }. c3 − a1 c2 = 0, d3 − a1 d2 = 0, b2,y − b1,y a1 − a1,y b1 = 0, IC 0 s : c1,x − c1,y + c1 d1 − c21 − b2 + a1 b1 = 0, d1,x − c1,y = 0. (2,2)

J3,9

(2,2)

J3,10 :

:

{z2 + a1 z1 , z1,y + b1 z1,x + b2 z1 , z1,xxx + c1 z1,xx + c2 z1,x + c3 z1 }.

b1,xx − 31 c1,y − 13 c1,x b1 + b2,x − 31 b1,x c1 = 0, c1,xy + c1,xx b1 + 6b2,xx − 3c2,y − 3c2,x b1 + 43 c1,y c1 + 2b1,x c1,x IC 0 s : + 34 c1,x b1 c1 + 2b2,x c1 − 6b1,x c2 + 43 b1,x c22 = 0, b2,xxx + b2,xx c1 − c3,y − c3,x b1 + b2,x c2 − 3b1,x c3 = 0. This explicit form provides a complete understanding of the features of a given system of linear homogeneous pde’s. Of particular importance is the clear separation of all correlations for the coefficients due to the coherence conditions given in the above theorem, and additional properties that may be due to the structure of a specific problem. For example, the lowest equation (2,2) in any Janet basis of type J3,5 has always the form z1,x + a2 z1 = 0, the vanishing of the term proportional to z2 is not a characteristic feature of a specific problem. The importance of this classification will become clear in the next chapter where Janet bases of these types occur as determining systems for certain symmetry groups. The quasilinearity of the constraints generated in step S5a (see page 59) allows application of the algorithm JanetBasis to the system C defined there with the following modification. The algorithm is interrupted if an equation is generated that is not quasilinear. It turns out that this does not occur for the types listed in Table 2.1, Table 2.2 or 2.3. The term ordering applied for the integrability conditions is always grlex with a1 < a2 < . . . < b1 < b2 < . . . < c1 < c2 < . . . and x < y. In most cases the Janet basis property is either obtained upon proper reordering or by autoreduction. (2,2)

Example 2.41 A typical case is type J3,3 the integrability conditions

for which in the first place

b1,x − 3a1,y = 0, a1,yy − 31 b2,x + 23 a1,y b1 = 0, a1,yyy + a1,yy b1 − b3,x + a1,y b2 = 0 are obtained. Autoreduction generates the Janet basis from it by replacing the third equation with b2,xy − 3b3,x + 13 b2,x b1 − 2b1,y a1,y + 3a1,y b2 − 23 a1,y b22 = 0.

68

Syzygies of Janet Bases. The coherence conditions of Theorems 2.15, 2.16 and 2.17 imply certain relations between the elements of a Janet basis which are known as syzygies. They will be required later on in this chapter for computing quotients. Let the Janet basis be f = {f1 , . . . , fp } where fi ∈ Dm , D = Q(x, y)[∂x , ∂y ]. Syzygies of f are relations of the form dk,1 f1 + . . . + dk,p fp = 0 where dk,i ∈ D, i = 1, . . . p, k = 1, 2, . . .. The (dk,1 , . . . , dk,p ) may be considered as elements of the module Dp . The totality of syzygies generates a submodule. Example 2.42 Consider the Janet basis {f1 ≡ zx + az, f2 ≡ zy + bz} with the constraint ay = bx . The coherence condition for zx − az − f1 = 0 and zy + bz − f2 = 0 yields azy + ay z − f1,y − bzx − bx z + f2,x = 0. Reduction w.r.t. to the given equations and some simplification yields the single syzygy (∂y + b)f1 − (∂x + a)f2 = 0. Example 2.43 

f1 ≡ zxx +

Consider the Janet basis

4 2 1 1 x 2 zx + 2 z, f2 ≡ zxy + zy , f3 ≡ zyy + zy − 2 zx − 2 z . x x y x y y

4 z + 2 z −f = 0 and z + 1 z −f = 0 The integrability condition for zxx + x x 1 xy x y 2 x2 3 f = 0. Similarly from yields upon reduction and simplification f1,y + f2,x − x 2 2 2 y y y y2 the last two elements f1 − x f2,y − x f2 + x f3,x + 2 f3 = 0 is obtained. x Autoreduction of these two equations yields the final answer
1 ∂ + 2 ∂ + 2
f = 0, ∂yy + y3 ∂y − x2 ∂x − 22 f2 − ∂xy + x y y x xy 3 y y f1 −

y2 y
y2 y2
∂ + f + ∂ + f3 = 0. y 2 x x x x x2

Gcrd and Lclm of Modules. The greatest common right divisor (Gcrd) or sum of two modules I ≡< f1 , . . . , fp > and J ≡< g1 , . . . , gq >, fi , gj ∈ Dm for all i and j, D = Q(x, y)[∂x , ∂y ], is generated by the union of the generators of I and J (Cox et al. [34], page 191). The solution space of the equations corresponding to Gcrd(I, J) is the intersection of the solution spaces of its arguments. From this definition the following algorithm is self-explanatory. Algorithm 2.9 Gcrd(I, J). Given two modules I ≡< f1 , . . . , fp > and J ≡< g1 , . . . , gq > in Dm , the Janet basis for its greatest common divisor is returned. S1 : Join input generators. Set h := {f1 , . . . , fp } ∪ {g1 , . . . , gq }

.

Linear Differential Equations

69

S2 : Create Janet basis. Transform h in a Janet basis and return the result. For m = 1 the Gcrd of the ideals I, J ∈ D is obtained. For m = n = 1 the Euclidean algorithm for computing the Gcrd of two linear ode’s is subsumed as a special case under the above algorithm. y 3 ∂ > and Example 2.44 Consider the ideals I =< ∂x + x ∂y , ∂xxx + x xx J =< ∂xx +

1 1 1 1 1 1 1 ∂x − 2 , ∂xy + ∂y + ∂x + , ∂yy + ∂y − 2 > . x x y xy y x y

Applying the above algorithm, the greatest common right divisor is obtained y as Gcrd(I, J) =< ∂x + x ∂y , ∂yy + y1 ∂y − 12 >. In terms of solution spaces y this result may be understood as follows. For I a basis of the solution space is  x y  1 x y 1, y , x , and for J a basis is xy , y , x . A basis for their two-dimensional  y intersection space is x y , x , it is the solution space of Gcrd(I, J). The Lclm of two modules I and J is the left intersection module of its arguments. The solution space of the equations corresponding to Lclm(I, J) is the smallest space containing the solution spaces of its arguments. An algorithm for generating the Lclm requires a more detailed analysis. Properly adjusting the corresponding algorithm from commutative algebra, see for example Cox et al. [34], Proposition 3.11 on page 218 and the article by Grigoriev and Schwarz [61], Section 3, the following algorithm is obtained. Algorithm 2.10 Lclm(I, J). Given the two modules I ≡< f1 , . . . , fp > and J ≡< g1 , . . . , gq > in Dm , the Janet basis for its least common left multiple is returned. S1 : Auxiliary system. Introducing two differential indeterminates w and z, generate the system {f1 w, . . . , fp w, g1 w − g1 z, . . . , gq w − gq z}. S2 : Generate Janet basis. For the system obtained in S1 generate a Janet basis in lex term order with w > z. S3 : Return result. From the Janet basis obtained in S2 collect elements depending on z only and return the respective operators generating the result. Example 2.45 Consider the two ideals 1 1 1 1 I =< ∂y + , ∂x + > and J =< ∂y + , ∂x + >. y x x+y x+y In step S1 the system 1 w, wy + y1 w, wx + x 1 w − z − 1 z, w + 1 w − z − 1 z wy + x + y x x y x+y x+y x+y

70 is generated. The Janet basis obtained in step S2 is zx −

x−y 2x + 4y y2 2 zy + z, zyy + zy + z, x2 x2 xy + y 2 xy + y 2 x2 + xy w+ zx + x y y z.

Consequently, the intersection ideal is Lclm(I, J) =< ∂x −

2x + 4y y2 x−y 2 ∂y + , ∂yy + ∂y + >. x2 x2 xy + y 2 xy + y 2

In some places later in this book a shortened notation for the Lclm is applied, e. g. Lclm(zy + az, zx + bz) = Lclm(∂y + a, ∂x + b)z. Exact Quotient and Relative Syzygies. For ordinary differential operators the exact quotient has been defined on page 22. In order to generate a full system of solutions for a system of linear pde’s which is reducible but not completely reducible, a proper generalization of this exact quotient is required. It is given next. Definition 2.9 (Exact quotient, relative syzygies) Let I ≡< f1 , . . . , fp >∈ Dm and J ≡< g1 , . . . , gq >∈ Dm be such that I ⊆ J. The exact quotient module is generated by {(ei,1 , . . . , ei,q ) ∈ Dq |ei,1 g1 + . . . + ei,q gq = fi , i = 1, . . . , p}. The relative syzygies module Syz(I, J) is generated by {h = (h1 , . . . , hq ) ∈ Dq |h1 g1 + . . . + hq gq ∈ I}.

If for two modules there holds I ⊆ J, J is also called a divisor or a component of I. Obviously the relative syzygies module extends the syzygy module of J, the latter is obtained for the special choice I = 0. The following algorithm generates the relative syzygies module as the sum of the exact quotient and the syzygies module. Algorithm 2.11 RelativeSyzygies(I, J). Given two modules I ⊂ J ⊂ Dm by I ≡< f1 , . . . , fp > and J ≡< g1 , . . . , gq >, the Janet basis for its relative syzygies module Syz(I, J) ∈ Dq is returned. S1 : Exact quotient. For i = 1, . . . , p determine ei = (ei,1 , . . . , ei,q ) from ei,1 g1 + . . . + ei,q gq = fi . S2 : Include syzygies. Generate the syzygies sk = sk,1 g1 + . . . + sk,q gq for k = 1, 2, . . .. S3 : Return Janet basis. Join the quotients qi and the syzygies sk , transform them into a Janet basis and return it.

Linear Differential Equations

Example 2.46

71

Consider the ideals

I =< ∂y −

2 2x + y x−y 2 x2 ∂x − , ∂xx + ∂x + > x x+y x(x + y) y2 y2

1 , ∂ + 1 >. There holds I ⊂ J. Step S1 generates the matrix and J =< ∂x + x y y   o n 2 3x + y x 1
− 2 , 1 , ∂x + , 0 . In step S2 the single syzygy ∂y + y1 , ∂x + x x(x + y) y is obtained. Step S3 generates the Janet basis for the relative syzygies module       2 3x + y x−y Syz(I, J) =< − x2 , 1 , ∂x + , 0 , ∂y − , 0 >⊂ D2 . x(x + y) y(x + y) y Example 2.47 (Li, Schwarz and Tsarev [105]) Consider the ideal I = xy − 1 y < ∂y − x y ∂x , ∂xx − x ∂x − x > and its right factor J =< ∂x − y, ∂y − x >. 
1
Division in step S1 yields −x y , 1 , ∂x + x , 0 . There is a single syzygy (−∂y + x, ∂x − y). From these three generators step S3 generates a Janet basis 1 for the relative syzygies module Syz(I, J) =< (− x y , 1), (∂x + x , 0), (∂y , 0) >. 1 ∂ , ∂ , ∂ + 1 ∂ >. There Example 2.48 Consider the ideal I =< ∂xx + x x xy yy y y is a single maximal ideal J =< ∂x , ∂y > containing it. It yields the relative 1 , 0
, (∂ , 0), (0, ∂ ), 0, ∂ + 1
>. It syzygies module Syz(I, J) =< ∂x + x y x y y may be represented as the intersection of two maximal modules of order 1, i.e., there holds
Syz(I, J) = Lclm < (1, 0), (0, ∂x ) 0, ∂y + y1 >,

1
< x y , 1 , ∂x + x , 0 , (∂y , 0) > . Similar as for ordinary differential equations there remains to be discussed how lower order right factors of a module of partial differential operators simplify the solution procedure, compare the discussion on page 31. The following result is due to Grigoriev and Schwarz [62]. Theorem 2.18 Let I ≡< f1 , . . . , fp > and J ≡< g1 , . . . , gq > be two submodules of Dm such that I ⊆ J, with solution space VI and VJ of finite dimension dI and dJ respectively. Let the relative syzygies module Syz(I, J) ≡ K ⊂ Dq be generated by < h1 , . . . , hr >, and z = (z1 , . . . , zm ) and z¯ = (¯ z1 , . . . , z¯q ) be differential indeterminates. A basis for the solution space VI may be constructed as follows. i) Determine a basis of VJ from g1 z = 0, . . . , gq z = 0, let its elements be vk = (vk,1 , . . . , vk,m ), k = 1, . . . , dJ .

72 ii) Determine a basis of VK from h1 z¯ = 0, . . . , hr z¯ = 0, let its elements be wk = (wk,1 , . . . , wk,q ), k = 1, . . . , dK = dI − dJ . iii) For each inhomogeneous system g1 z¯ = wk,1 , . . . , gq z¯ = wk,q , k = 1, . . . , dK determine a special solution v¯k = (¯ vk,1 , . . . , v¯k,m ). A basis of the solution space VI is {v1 , . . . , vdJ , v¯1 , . . . , v¯dK }. Example 2.49 In Example 2.46, the right factor J corresponds to the 1 z = 0, z + 1 z = 0, its solution yields the basis element system zx + x y y 1 . The relative syzygies lead to the system v1 = xy z2 −

x2 3x + y x−y 2 z1 = 0, z1,x + x(x + y) z1 = 0, z1,y − y(x + y) z1 = 0, y

y x , z2 = . Substituting it x(x + y)2 y(x + y)2 1 . into the inhomogeneous system results in the special solution v¯1 = x + y  1 1 Consequently, a basis for the originally given system is xy , x + . y

it yields the basis element z1 =

Example 2.50 In Example 2.47 the system of pde’s corresponding to J is zx − yz = 0, zy − xz = 0, it yields the basis {exy } for VJ . The relative 1 syzygies module leads to the system z2 − x y z1 = 0, z1,x + x z1 = 0 and 1, z = 1. z1,y = 0; its one-dimensional solution space is generated by z1 = x 2 y 1 , z − xz = 1 with There is a single inhomogeneous system zx − yz = x y y the special solution v¯1 = exy Ei(−xy). The exponential integral is defined R by Ei(ax) = eax dx x . A basis for the solution space of the originally given xy xy system is {e , e Ei(−xy)}. Example 2.51 In Example 2.48, to the first order right factor corresponds the system zx = 0, zy = 0, it yields the basis {1} vor VJ . The two arguments of the Lclm lead to the systems z¯1 = 0, z¯2,x = 0, z¯2,y + y1 z¯2 = 0 with the 1 solution z¯1,1 = 0, z¯1,2 = 0, and z¯2 + x y z¯1 = 0, z¯1,x + x z¯1 = 0, z¯1,y = 0 with 1 , z¯ = − 1 . Substituting them into the inhomogeneous the solution z¯2,1 = x 2,2 y system yields the two special solutions log y and log x y . Therefore a basis for  x the given system is 1, log y, log y . For efficiency reasons it is advantageous to have closed forms available for those relative syzygies that occur frequently in applications. In this way the Janet basis calculation which occurs in step S3 of the above algorithm may be avoided. Because these applications usually involve solving differential equations, they are given in terms of differential polynomials.

Linear Differential Equations

73

Lemma 2.4 A Janet basis {zy + A1 zx + A2 z, zxx + B1 zx + B2 z} of type (1,2) (1,2) J2,1 has a component {z1 ≡ zx + az, z2 ≡ zy + bz} of type J1 iff there holds b − A2 + A1 a = 0, ax − a2 + B1 a − B2 = 0, ay + A1 a2 + (A1,x − A1 B1 )a − A2,x + A1 B2 = 0. (2,2)

The relative syzygies module is of type J1,2

and is explicitly given by

{z2 + A1 z1 , z1,x + (B1 − a)z1 , z1,y + (A1,x − A1 B1 + 2aA1 + b)z1 }.

(1,2)

Proof Reduction of the type J2,1 Janet basis w.r.t. z1 ≡ zx + az and z2 ≡ zy + bz yields the first two generators of the quotient under the assumption that b is determined from the first constraint and the equation with leading term ax is valid. The third element defining the quotient follows if in the integrability condition ay −bx = 0 the variable b is eliminated applying the first two constraints. In Example 2.42 it has been shown that the type (1,2) J1 Janet basis obeys the syzygy z1,y − z2,x + bz1 − az2 = 0. Reducing it w.r.t. to the two previous relations, the last generator is obtained. Example 2.52 Consider again the Janet bases from Example 2.46. By the above lemma, the coefficients A1 = −

x2 x−y 2 2x + y 2 2 , A2 = − 2 , B1 = x x + y and B2 = x(x + y) y y

yield immediately the Janet basis for the relative syzygies module       2 3x + y x−y < − x2 , 1 , ∂x + , 0 , ∂y − ,0 > . x(x + y) y(x + y) y Lemma 2.5

(1,2)

A Janet basis {zx + A1 z, zyy + B1 zy + B2 z} of type J2,2

a component {z1 ≡ zx + az, z2 ≡ zy + bz} of type

(1,2) J1

has

iff there holds

A1 − a = 0, by − b2 − B1 b − B2 = 0, bx + ay − B1,x = 0. (2,2)

Its relative syzygies module is of type J1,1

and is explicitly given by

{z1 , z2,x + az2 , z2,y + (B1 − b)z2 }.

The proof is similar as for Lemma 2.4 and is therefore omitted.  1 z, z + 1 z − 1 z . Example 2.53 Consider the Janet basis zx + x yy y y y2   1 1 1 Dividing out z1 ≡ zx + x z, z2 ≡ zy − y z yields z1 , z2,x + x z2 , z2,y + y2 z2 . y 1 y The original system has the solutions x , xy whereas x is the only solution

74 of its first order component {z1 , z2 }. The second member is obtained by solving the quotient system with the result z1 = 0, z2 = − 2 2 , and then the xy 1 z = 0, z − 1 z = − 2 with the result 1 . inhomogeneous system zx + x y y xy xy 2 Explicit forms for the possible quotients of Janet bases of order three are given next without proof. Lemma 2.6 If a Janet basis {zy + A1 zx + A2 z, zxxx + B1 zxx + B2 zx + B3 z} (1,2) (1,2) of type J3,1 has a component {z1 ≡ zx + az, z2 ≡ zy + bz} of type J1 , (2,2)

its quotient is of type J2,4 

and is explicitly given by

z2 + A1 z1 , z1,y + A1 z1,x + (A1,x + A2 )z1 , z1,xx + (B1 − a)z1,x + (B2 − aB1 − 2ax + a2 )z1 .

If it has a component {z1 ≡ zy + a1 zx + a2 z, z2 ≡ zxx + b1 zx + b2 z} of type (1,2) (2,2) J2,1 , its quotient is of type J1,1 and is explicitly given by 

z1 , z2,x + (B1 − b1 )z2 , z2,y + (2A1,x − A1 B1 + b1 A1 + A2 )z2 .

 3 Consider the Janet basis zy + x y zx , zxxx + x zxx . Dividing  (1,2) 1 z, z + 1 z yields the type J (2,2) Janet out the type J1 component zx − x y 1,1 y basis x 4 1 2 {z2 + x y z1 , z1,y + y z1,x + y z1 , z1,xx + x z1,x + x2 z1 . Lemma 2.7 If a Janet basis  zxx + A1 zy + A2 zx + A3 z, zxy + B1 zy + B2 zx + B3 z, zyy + C1 zy + C2 zx + C3 z Example 2.54

(1,2)

of type J3,2

(1,2)

has a component {z1 ≡ zx + az, z2 ≡ zy + bz} of type J1

its quotient is of type 

(2,2) J2,3

,

and is explicitly given by

z1,x + A1 z2 + (A2 − a)z1 , z1,y + (B1 − a)z2 + B2 z1 , z2,x + B1 z2 + (B2 − b)z1 , z2,y + (C1 − b)z2 + C2 z1 .

If it has a component {z1 ≡ zy + a1 zx + a2 z, z2 ≡ zxx + b1 zx + b2 z} of type (1,2) (2,2) J2,1 , its quotient is of type J1,2 and is explicitly given by {z2 + A1 z1 , z1,x + (a1 A1 + B1 )z1 , z1,y + (C1 − 2a21 A1 − a2 )z1 }.

Linear Differential Equations

75 (1,2)

If it has a component {z1 ≡ zx + a1 z, z2 ≡ zyy + b1 zy + b2 z} of type J2,2 , its (2,2)

quotient is of type J1,2

and is explicitly given by

{z2 + C2 z1 , z1,x + (A2 − a1 )z1 , z1,y + B2 z1 }. Example 2.55

(1,2)

Consider the type J3,2

Janet basis

n 2 1 1 x 2 o 4 zxx + zx + 2 z, zxy + zy , zyy + zy − 2 zx − 2 z . x x y x y y (1,2)

Dividing out the type J1 (2,2)

the type J2,3

 1 z, z ≡ z + 1 z yields Janet basis z1 ≡ zx + x 2 y y

Janet basis

n 3 1 1 x o z1,x + z1 , z1,y , z2,x + z2 − z1 , z2,y − 2 z1 . x x y y 1 z, z ≡ z + 1 z − 1 z} Janet basis {z1 ≡ zx + x 2 yy y y y2  x 3 Janet basis z2 − 2 z1 , z1,x + x z1 , z1,y . y (1,2)

Dividing out the type J2,2 (2,2)

yields the type J1,2

Lemma 2.8 If a Janet basis {zx + A1 z, zyyy + B1 zyy + B2 zy + B3 z} of (1,2) (1,2) type J3,3 has a component {z1 ≡ zx + az, z2 ≡ zy + bz} of type J1 , its (2,2)

quotient is of type J2,2 

and is explicitly given by

z1 , z2,x + A1 z2 , z2,yy + (B1 − b)z2,y + (B2 − bB1 − 2by + b2 )z2 . (1,2)

If it has a component {z1 ≡ zx + a1 z, z2 ≡ zyy + b1 zy + b2 z} of type J2,2 , its (2,2)

quotient is of type J1,1

and is explicitly given by

{z1 , z2,x + A1 z2 , z2,y + (B1 − b1 )z2 }. Construction of Janet Bases. It is difficult to construct examples for the various types of Janet bases by prescribing the coefficients. The reason is that finding sets of coefficients satisfying the coherence conditions comes down to solving systems of nonlinear pde’s for which solution algorithms are not known. A different approach starts from the solutions. A basis for the solution space is given, and the problem is to generate a Janet basis with this solution space. For a single linear ode the answer has been given in (2.4), providing explicit expressions for the coefficients in terms of a fundamental system. For the two second order Janet bases of type J (1,2) the answer is to be obtained in Exercises 2.9 and 2.10. It should be emphasized, however, that proceeding in this way only special classes of Janet bases may be obtained, i.e., those that are completely

76 reducible with first order right factors. Due to the requirement that the Janet basis to be constructed has the rational function field of the independent variables as base field, the fundamental system has to be hyperexponential. Any other choice in general will lead to coefficients that are not in this base field. This problem exists already for linear ode’s. An algorithm will be described now which is based on the constructive proof of a theorem due to Lie [112], vol. I, page 183. In the special case where the dependency on the integration constants is linear, the desired system is also linear and homogeneous. Assume the following system of equations is given. z1 ≡ C1 z1,1 + . . . + Cr z1,r , .. .. . .

(2.36)

zm ≡ C1 zm,1 + . . . + Cr zm,r where zj,k , j = 1, . . . , m, k = 1, . . . , r are given functions of x and y, z1 , . . . , zm are undetermined functions of x and y, and the Ck are constants to be eliminated. To this end, by a suitable number of differentiations, an enlarged system of rank r is generated from which the Ck may be eliminated. Upon resubstitution, the desired system of pde’s is obtained. The algorithm ConstructJanetBasis takes expressions (2.36) as input and returns the corresponding Janet basis in the specified term ordering. Algorithm 2.12 ConstructJanetBasis({z1 , . . . zm }, O). Given a system of expressions of the form (2.36), the Janet basis in the term ordering O is returned such that (2.36) is its general solution. S1 : Set up linear system. Determine d such that (d + 1)(d + 2) > r. S2 : Rank? Generate linear system (L) for C1 , . . . , Cr by differentiating w.r.t. x and y up to order d. It has the form with j = 1, . . . , m. C1 zj,1 + . . . + Cr zj,r = zj , C1 zj,1,x + . . . + Cr zj,r,x = zj,x , C1 zj,1,y + . . . + Cr zj,r,y = zj,y , .. .. .. . . . C1 zj,1,yy... + . . . + Cr zj,r,yy... = zj,yy... ,

(L)

S3 : Rank? If the rank of the coefficient matrix is less than r, set d := d+1 and goto S2. S4 : Determine C’s. Take a subsystem of (L) of rank r and determine C1 , . . . , Cr from it. S5 : Generate Janet basis. Substitute the values for the C’s obtained in step S4 into the remaining equations and generate a Janet basis. If its solution space has dimension r, return it, otherwise set d := d + 1 and goto S2.

Linear Differential Equations

77

Example 2.56 For m = 1, r = 2 let z1 = z = C1 (x2 − y 2 ) + C2 xy. In step S1, d = 1 is found. Step S2 generates the system of equations C1 (x2 − y 2 ) + C2 xy = z, 2C1 x + C2 y = zx , −2C1 y + C2 x = zy . The rank of the coefficient matrix is two. In step S4 the last two equations are employed for determining C1 and C2 with the result C1 =

y x zx − zy , 2(x2 + y 2 ) 2(x2 + y 2 )

C2 =

y x zx + 2 zy . x2 + y 2 x + y2

Substitution into the first equation yields a single first order equation for z. Therefore in step S5 the value of d is increased to 2 and the algorithm continues with step S2 where the system 2C1 = zxx , C2 = zxy , −2C1 = zyy is generated. Substituting C1 and C2 and joining the previously obtained first order equation leads in step S5 to the Janet basis zx +

y 2y 2 2 zy − z = 0, zyy − 2 zy + 2 z=0 x x x + y2 x + y2

(2.37)

which is the final answer.

2.4

Solving Partial Differential Equations

Until now the goal has been to obtain information on the solutions of linear pde’s without determining them explicitly. Furthermore, it has been shown how given pde’s may be replaced by simpler ones, e. g. by decomposing them into lower order equations. A common feature of these efforts was that the necessary calculations took their course within the base field of the originally given equations. In this section, the goal is to determine solutions in closed form. This may require to enlarge the function field. Various cases corresponding to special values of n and m, the number of independent and dependent variables respectively, and the number N of equations will be considered. Except in the last subsection, all equations are of first order. Linear Equations for a Single Function. First order systems containing a single dependent variable and any number of independent variables, i.e., m = 1 and n arbitrary, have been studied extensively in the literature. The discussion in this section follows closely the Encyklop¨ adie article by von Weber [187], or Chapter II in Goursat’s book [57], see also Kamke [87], page 45 ff. At first, N = 1 is assumed. Let the function z(x1 , . . . , xn ) be determined by the homogeneous equation Az ≡ a1

∂z ∂z ∂z + a2 + . . . + an = 0. ∂x1 ∂x2 ∂xn

(2.38)

78 In particular there is no term proportional to z, and ak ≡ ak (x1 , . . . , xn ) belongs to Q(x1 , x2 , . . . , xn ) or an extension of it. A fundamental system or integral basis for this equation is a set of n − 1 independent functions ψ1 (x1 , . . . , xn ), . . . , ψn−1 (x1 , . . . , xn ) such that Aψk = 0 for k = 1, . . . , n − 1. The following theorem reduces the solution of (2.38) to solving a system of ode’s. Its proof may be found in the literature quoted above. The general solution of equation (2.38) may be described

Theorem 2.19 as follows.

i) A fundamental system for (2.38) is given by n-1 first integrals of the so-called characteristic system dxn dx1 = a1 (x1 , . . . , xn ), . . . , = an (x1 , . . . , xn ). dt dt ii) The general solution of (2.38) is given by an undetermined function Ψ(ψ1 , . . . , ψn−1 ) of the first integrals. It should be noted, however, that there is no guarantee that the fundamental system may always be explicitly determined. The reason is that solving the characteristic system amounts to solving general first order ode’s for which an algorithm does not exist. There is a remarkable generalization of the above theorem to inhomogeneous quasilinear equations by Kamke [86] II, §3, no. 15. A solution of the quasilinear first order equation

Theorem 2.20

a1

∂z ∂z ∂z + a2 + . . . + an =r ∂x1 ∂x2 ∂xn

(2.39)

where ak ≡ ak (x1 , . . . , xn , z) for k = 1, . . . , n and r ≡ r(x1 , . . . , xn , z) may be obtained as follows. Determine any integral w = ψ(x1 , . . . , xn , z) of a1

∂w ∂w ∂w ∂w + a2 + . . . + an +r =0 ∂x1 ∂x2 ∂xn ∂z

by Theorem 2.19 and solve ψ(x1 , . . . , xn , z) = C for z; C 6= 0 is a constant. The special case n = 2 of the above theorem will occur frequently later on in this book. Therefore it is described in more detail in the following corollary. Corollary 2.7 tion

The general solution of the linear partial differential equazx + a1 zy + a2 z + a3 = 0

(2.40)

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79

with ak ≡ ak (x, y) ∈ Q(x, y), k = 1, . . . , 3 may be described as follows. Let φ(x, y) = const be the first integral obtained from the solution of the first order dy ¯ y¯) = a1 (x, y). Let y¯ = φ(x, y) and assume that the inverse y = φ(x, ode dx exists. Then the general solution of (2.40) is Z Z Z 

¯ exp ¯ ¯ z(x, y) = Φ(φ)− a3 (x, φ) a2 (x, φ)dx dx exp − a2 (x, φ)dx

y¯=φ

(2.41) with Φ an undetermined function of its argument. Proof It is based on the procedure described in Kamke [87], Section 4.2. A homogeneous equation corresponding to (2.40) for a new function w(x, y, z) is wx + a1 wy − (a2 z + a3 )wz = 0. A first integral φ is obtained dy from = a1 . If it is applied for introducing a new variable y¯, the equation dx w ¯x − (a2 z + a3 )|y=φ¯w ¯z = 0 for w ¯ is obtained with the first integral Z Z Z  ¯ ¯ exp ¯ ψ = z exp a2 (x, φ)dx + a3 (x, φ) a2 (x, φ)dx dx. Consequently, the general solution w = Ψ(φ, ψ) is obtained with an undetermined function Ψ. Resubstituting y¯ = φ(x, y) and solving for z yields (2.41). This corollary is particularly useful if a special solution of the homogeneous equation is already known, then solving the inhomogeneous equation is reduced to an integration. Let 2xzx + 3yzy = 1 be the given equation. With the 3y 1 . The first order notation of Corollary 2.7, a1 = 2x , a2 = 0 and a3 = − 2x 2 dy 3y y equation = 2x leads to φ(x, y) = 3 . From (2.41) the general solution is dx x  2  y − 21 log x with Φ an undetermined obtained in the form z(x, y) = Φ x3 function of its argument. Example 2.57

Now, let N > 1. It is interesting to see how several concepts like integrability conditions and reductions have evolved in the course of time for this particular case. The subsequent theorem shows how the Jacobian normal form (Kamke [87], Chapter 6.2) is obtained naturally as a Janet basis. Theorem 2.21 given as

Let a first order system for a function z(x1 , . . . , xn ) be a1,1 ∂z + a1,2 ∂z + . . . + a1,n ∂z = 0, ∂x1 ∂x2 ∂xn ∂z ∂z a2,1 + a2,2 + . . . + a2,n ∂z = 0, ∂x1 ∂x2 ∂xn ... ... ak,1 ∂z + ak,2 ∂z + . . . + ak,n ∂z = 0. ∂x1 ∂x2 ∂xn

(2.42)

80 The coefficients ai,j (x1 , . . . , xn ) belong to some function field F. For the variable ordering x1 > x2 > . . . > xn , the Janet basis for this system is identical to the Jacobian normal form, possibly after a new enumeration of the xi , ∂z + . . . + b ∂z = 0, ∂z + b r,r+1 r,n ∂xr ∂xr+1 ∂xn ∂z + . . . + b ∂z = 0, ∂z + b r−1,r+1 r−1,n ∂xr−1 ∂xr+1 ∂xn ...

(2.43)

∂z + b ∂z + . . . + b ∂z = 0 1,r+1 1,n ∂x1 ∂xr+1 ∂xn with 1 ≤ r ≤ n and bi,j ∈ F. Proof By algebraic reduction of the system (2.42), the following autoreduced form may always be achieved, possibly after a new enumeration of the independent variables. ∂z + a ¯p,p+1 ∂z + . . . + a ¯p,n ∂z = 0, ∂xp ∂xp+1 ∂xn ∂z + a ∂z ¯p−1,p+1 + ... + a ¯p−1,n ∂z = 0, ∂xp−1 ∂xp+1 ∂xn ... ...

(2.44)

∂z + a ¯1,p+1 ∂z + . . . + a ¯1,n ∂z = 0 ∂x1 ∂xp+1 ∂xn with 1 ≤ p ≤ n. The integrability conditions have the form   ∂ ∂z + a ¯i,p+1 ∂z + . . . + a ¯i,n ∂z ∂xj ∂xi ∂xp+1 ∂xn   ∂ ∂z ∂z − +a ¯j,p+1 + ... + a ¯j,n ∂z = 0 ∂xi ∂xj ∂xp+1 ∂xn for 1 ≤ i < j ≤ p. All second order terms in these expressions cancel each other identically. The remaining first order equations are reduced w.r.t. to (2.44). If the result is always zero, (2.44) is the desired Janet basis. Otherwise those that do not vanish are added to the system (2.44) and the process is repeated. The results described in the preceding section guarantee that this process terminates after a finite number of iterations. Example 2.58

The following system of pde’s

zx2 + (x4 x3 − x2 )zx3 + (x4 x3 x1 − x2 x1 + x2 )zx4 = 0, zx1 + (x4 + x2 − 3x1 )zx3 + (x4 x1 + x3 + x2 x1 )zx4 = 0 for a function z ≡ z(x1 , x2 , x3 , x4 ) has been discussed by Imschenetzky [79]; as usual zxi ≡ ∂z . The Janet basis for x1 > x2 > x3 > x4 is ∂xi  zx3 + x1 zx4 , zx2 + x2 zx4 , zx1 + (x3 + 3x21 )zx4

Linear Differential Equations

81

and for x4 > x3 > x2 > x1 it is o n x1 1 x2 z , z − z , z + z . zx2 − x x x x x 1 3 1 4 1 x3 + 3x21 x3 + 3x21 x3 + 3x21 From each of these Janet bases it is obvious that the general solution depends on an undetermined function of a single argument. Explicitly it is given by
φ x4 − x1 x3 − 12 x22 − x31 . Linear Equations with Symmetries. In some applications it is advantageous to transform a linear equation for a function f (x1 , x1 , . . . , xn ) Af ≡ a1

∂f ∂f ∂f + a2 + . . . + an =0 ∂x1 ∂x2 ∂xn

(2.45)

to a new set of variables y1 , . . . , yn . Let them be defined by yk ≡ ψk (x1 , . . . , xn ) with the inversions xk ≡ φk (y1 , . . . , yn ). In these variables (2.45) becomes B1

∂g ∂g ∂g + B2 + . . . + Bn =0 ∂y1 ∂y2 ∂yn

(2.46)

where Bk = Aψk |x1 =φ1 ,...,xn =φn for 1 ≤ k ≤ n and g ≡ f (φ1 , . . . , φn ). The proof of this expression is considered in Exercise 2.21. Let A = x∂x + y∂y by given and q new variables u and v be √ v introduced by x = uv, y = u with the inverse u = x y , v = xy. Applying the above rule yields the transformed operator v∂v . Example 2.59

Of particular interest are those nontrivial transformations leaving the equation Af = 0 unchanged. Due to its importance a special term is introduced for them. Definition 2.10 (Symmetry of a linear pde) If a linear homogeneous pde (2.45) is invariant w.r.t. a variable transformation, it is called a symmetry of (2.45). This is also expressed by saying that the transformation is admitted by (2.45). It may occur that the totality of transformations admitted by a given equation (2.45) make up a group depending on one or more parameters. For this concept the reader is referred to Section 3.1. A convenient criterion for the invariance w.r.t. a group may be given in terms of its infinitesimal generators as follows. Theorem 2.22 A linear homogeneous pde (2.45) is invariant under the group with the infinitesimal generator X = ξ1 ∂ + . . . + ξn ∂ if and only ∂x1 ∂xn if there holds [A, X] = ρ(x1 , . . . , xn )A. The commutator [A, X] is defined by equation (3.10) in the next chapter. The solution procedure for an equation (2.45) varies with the type of the

82 symmetry group that it admits. A detailed discussion may be found in the original work of Lie [108], § 10. Two special cases that are of particular importance later on in this book are described next. The first one deals with a solvable symmetry group, for its definition see page 118. Theorem 2.23 (Lie 1877) Let the linear homogeneous pde to be solved be given in the form Af ≡ a1

∂f ∂f ∂f + a2 + . . . + an = 0. ∂x1 ∂x2 ∂xn

Assume that it admits Xk ≡ ξk,1

∂ ∂ ∂ + ξk,2 + . . . + ξk,n ∂x1 ∂x2 ∂xn

Pk−1 ρ for k = 1, . . . , n − 1, and that the relations [Xi , Xk ] = ρ=1 ci,k Xρ for i < k and i = 1, . . . , n − 1, k = 2, . . . , n − 1, cρi,k constant are valid. Then a fundamental system may be obtained as follows. Let the determinant ∆ be defined by a1 a2 . . . an ξ1,1 ξ1,2 . . . ξ1,n ∆= . .. 6= 0. .. . ξn−1,1 ξn−1,2 . . . ξn−1,n Then an integral Φ1 of Af = 0 is dx1 dx2 a1 a2 Z 1 ξ1,1 ξ1,2 Φ1 = ∆ .. . ξn−2,1 ξn−2,2

. . . . ξn−2,n

. . . dxn . . . an . . . ξ1,n .. .

(2.47)

If new variables x ¯1 = Φ1 , x ¯k = xk for k = 2, . . . , n are introduced, derivatives ¯ = 0 and X ¯ k f = 0 for k = 1, . . . , n − 3 w.r.t. to x ¯1 do not occur any more, Af ¯ are a complete system allowing Xn−2 . This problem is of the same type as the original one with the number of variables diminished by one for which the same procedure may be repeated until n − 1 integrals are obtained. This theorem will be employed in Chapter 7 for solving ode’s with symmetries. In order to apply it, the knowledge of a sufficient number of symmetries of a certain kind is required. Because there is no algorithmic method for determining them, this knowledge must come from the application from which the equation Af = 0 originates. There are other methods for taking advantage of symmetries for solving first order pde’s or systems of pde’s. A few cases are described in the subsequent examples.

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83

y y Example 2.60 Let A ≡ ∂x + z ∂y + z ∂z , X1 ≡ ∂x , X2 ≡ x∂x + y∂y + z∂z . Due to [X1 , A] = 0, [X2 , A] = −A, [X1 , X2 ] = X1 the above theorem can be y applied. In the first step ∆ = x (y − z) and the integral for (2.47) is Z

y  z 1 y dy − dz = 2 log (y − z). yy−z z z

Therefore Φ1 = y − z may be chosen. Introducing new variables x ¯ = x, ¯ 1 = ∂x¯ . Integration and resubstitution y¯ = y − z, z¯ = z yields A¯ = x ¯∂x¯ and X gives Φ2 = x − z + (y − z) log y. Theorem 2.24 (Lie 1874) Let the linear homogeneous pde Af = 0 as in the preceding theorem allow q infinitesimal generators B1 , . . . , Bq . Assume that there are exactly m relations of the form k Bk = βm+1 Bm+1 + . . . + βqk Bq + αA,

k = 1, . . . , m.

Then the coefficients βjk obey Aβjk = 0, i.e., they are integrals of the given linear pde Af = 0. The proof may be found in Lie[107], §8, Theorem VII. A simple application is given next. z(z 2 + 1) Example 2.61 Let A = ∂x + z∂y − ∂z . It allows the three 2x symmetry generators U1 = ∂y , U2 = x∂x +y∂y and U3 = xy∂x + 21 y 2 ∂y −xz 2 ∂z because [A, U1 ] = 0, [A, U2 ] = A, [A, U3 ] = (xz + y)A. The relation U3 +

 2xz   2xz(xz − y) y 2  2x2 z − y U + + U − A=0 2 1 2 z2 + 1 z2 + 1 z2 + 1

2xz(xz − y) y 2 yields the integrals Φ1 = 22xz − y and Φ2 = + 2. z +1 z2 + 1 Linear Systems for m=1, n=2 and N=2. In this subsection various kinds of linear pde’s are considered that will occur frequently in the applications later on in this book. Usually its solution may be expressed in terms of path integrals in the x − y−plane. If p ≡ p(x, y) and q ≡ q(x, y) are two functions of x and y obeying py = qx , it is defined by I Z x Z y Z y Z x pdx + qdy = p(ξ, y0 )dξ + q(x, η)dη = q(x0 , η)dη + p(ξ, y)dξ. x0

y0

y0

x0

(2.48) The lower integration limit of the first integral contributes the integration constant. As a rule, any integration problem will be considered as solved as soon as it has been reduced to a form that is covered by the lemmata of this subsection. A straightforward generalization of a linear homogeneous ode is considered first.

84 Lemma 2.9

The general solution of the homogeneous system zx + az = 0,

zy + bz = 0

H where a ≡ a(x, y), b ≡ b(x, y) and ay = bx is z = exp (− adx + bdy). Example 2.62 Consider zx + xy 2 z = 0 and zy + x2 yz = 0. The integral (2.48) yields I Z Z xy 2 dx + x2 ydy = ξy02 dξ + x2 ηdη = 21 y02 (x2 − x20 ) + 12 x2 (y 2 − y02 ) = 12 x2 y 2 + C where C is a constant, and finally z = C exp (− 12 x2 y 2 ). Lemma 2.10

The general solution of the inhomogeneous system zx + az = p,

zy + bz = q

where a, b, p and q depend on x and y and obey the integrability conditions ay = bx and py − qx = aq − bp may be written in the form   I pdx + qdy  I  + C where z0 = exp − adx + bdy z = z0 z0 is the solution of the homogeneous system with p = q = 0 and C is a constant. Proof Assume that the desired solution has the form z = C(x, y)z0 with z0 a solution of the homogeneous system. Differentiation and substitution p q into the given system leads to Cx = z0 and Cy = z0 from which the given result is obvious. Example 2.63 Consider the system zx + yz = y, zx + xz = x. By Lemma 2.9 the homogeneous system has the solution z0 = Ce−xy . Substituting it into the integral of the preceding lemma leads to I  −xy z=e exy (ydx + xdy) + C = Ce−xy + 1. (1,2)

A special second order system corresponding to Janet basis type J2,1 will occur frequently later on is treated next.

that

Lemma 2.11 Let the function z(x, y) be determined by the system of equations zy + azx = 0, zxx + czx = 0 with (ax − ac)x = cy . The nonconstant member of a fundamental system is given by I I
z = exp − cdx + (ax − ac)dy (dx − ady).

Linear Differential Equations

85

The proof of this lemma is the subject of Exercise 2.27. An example of such a system is

Example 2.64 zy +

x2 − 2x + 2 x zx = 0, zxx − zx = 0. (x − 1)y x(x − 1)

Applying the formulas given above yields for the nonconstant member of a 1 ex . In this particular case the same answer may fundamental system z = xy  1 1 be obtained from the component zx − x − x z, zy + y z because the given system is completely reducible. In general, however, this is not true and the expression in the lemma must be applied. Quasilinear Systems for m=1, n=2 and N≥2. Systems comprising more than two equations are the next topic. Moreover they do not need to be linear in the dependent variable any more. Lemma 2.12

Let a system of first order quasilinear equations ai zx + bi zy = ri ,

2≤i≤N

for z ≡ z(x, y) be given where ai ≡ ai (x, y), bi ≡ bi (x, y) and ri ≡ ri (x, y, z), a1 b2 − a2 b1 6= 0. Then z obeys zx = − b 1 r 2 − b 2 r 1 , a1 b2 − a2 b1

zy = a1 r2 − a2 r1 , a1 b2 − a2 b1

(a1 b2 − a2 b1 )rj + (a2 bj − aj b2 )r1 − (a1 bj − aj b1 )r2 = 0 f or j ≥ 3. (2.49) Proof The expressions for zx and zy follow by algebraic elimination from the first two equations. The remaining conditions either assure the coherence of the full system, or they reduce the originally given differential equations to an elimination problem. Depending on the form of the right hand sides ri , various alternatives have to be distinguished. If the ri do not contain z, or if only r1 and r2 depend linearly on z and the remaining conditions are identically satisfied, the first two equations lead to a representation of z as a path integral by Lemma 2.9 or 2.10. If any ri contains z, and if the conditions for j ≥ 3 given in the above lemma may be solved for z, the general solution does not contain a constant, and the first two equations reduce to consistency conditions. If they are violated the full system is inconsistent. Systems of this kind occur in the next chapter when groups acting on a two-dimensional manifold are transformed into canonical form. Partial Riccati-Like Systems: A Single Function. The decomposition of Janet bases into irreducible components has been shown to come down to

86 solving various kinds of partial differential equations. Their structure suggests to call them partial Riccati-like equations. In this section, those systems occuring in the decomposition of Janet bases of type J (1,2) of order not higher than three will be considered. First of all a few definitions are introduced. Two rational functions f, g ∈ Q(x, y) are defined to be an integrable pair (f, g) if ∂y f = ∂x g. Two integrable pairs (f, g) and (p, q) are said to be equivalent, denoted by ” ∼ ”, if there exists ∂ h a nonzero h ∈ Q(x, y) such that f − p = ∂x h and g − q = y ; ∼ defines an h h equivalence relation on the set of integrable pairs. Let a and b belong to Q(x, y). If they are not an integrable pair it may be possible that there exist two polynomials p, q ∈ Q[x, y] such that ∂ q
∂ p is an integrable pair. In the above mentioned article a + px , b + qy by Li and Schwarz [104] an algorithm is described that allows determining a pair (p, q) for any given (a, b) if it does exist. An element a of some larger function field comprising Q(x, y) is said to be x a and ∂y a belong to Q(x, y). If a is a hyperexa hyperexponential if both ∂a a ∂y a
∂ a x ponential, a , a is an integrable pair. Conversely, for an integrable pair H (f, g), the hyperexponential exp ( f dx + gdy) may be constructed. If (a, b) and (p, q) are two integrable pairs, then (f, g) ∼ (p, q) if and only if the ratio of their hyperexponentials is in Q(x, y), possibly multiplied by some constant. The simplest Riccati-like system involving a single unknown function is the subject of the next theorem. Theorem 2.25

The first order Riccati-like system of pde’s

e1 ≡ zx + A1 z 2 + A2 z + A3 = 0, e2 ≡ zy + B1 z 2 + B2 z + B3 = 0

(2.50)

is coherent and its general solution depends on a single constant if and only if its coefficients satisfy the constraints A1,y − B1,x − A1 B2 + A2 B1 = 0, A2,y − B2,x − 2A1 B3 + 2B1 A3 = 0,

(2.51)

A3,y − B3,x − A2 B3 + A3 B2 = 0. Let Ak , Bk ∈ Q(x, y), A1 6= 0, B1 6= 0 and system (2.51) be satsified. If (2.50) has a rational solution, one of the following alternatives applies. i) The general solution is rational and has the form 1 rx 1 ry +p= +p A1 r + C B1 r + C

(2.52)

where p, r ∈ Q(x, y) and C is the integration constant. ii) There is a single one, or there are two inequivalent special rational solutions.

Linear Differential Equations

87

Proof The coefficient constraints are the integrability conditions for sysu into the given equations yields tem (2.50). Substituing z = w ux +

u(A1 u − wx ) u(B1 u − wy ) + A2 u + A3 w = 0, uy + + B2 u + B3 w = 0. w w

In order to obtain a linear system for u and w, the constraints wx − A1 u + P w = 0, wy − B1 u + Qw = 0

(2.53)

are necessary where P, Q ∈ Q(x, y) still have to be determined. Substituting these conditions into the above system for u and w yields ux + (A2 + P )u + A3 w = 0, uy + (B2 + Q)u + B3 w = 0.

(2.54)

The consistency of the above substitution requires that the system comprising the equations (2.53) and (2.54) be a Janet basis. This is assured if Py − Qx + A1 B3 − A3 B1 = Py − Qx + 21 (A2,y − B2,x ) = 0. The second form is obtained by applying (2.51). A possible choice for P and Q is P = − 12 A2 , Q = − 12 B2 respectively. It leads to the linear system ux + 21 A2 u + A3 w = 0, uy + 21 B2 u + B3 w = 0, wx − A1 u − 21 A2 w = 0, wy − B1 u − 21 B2 w = 0. Let u = C1 u1 + C2 u2 , w = C1 w1 + C2 w2 be its general solution. If the ratio u w is rational, it may be rewritten as u = z= w Applying

u1 u2 u2 u1 w2 + − u1 w1 w2 w1 w1 w1 + . w2 = w2 w 1 C+ C+ w1 w1

C

   w2,x w1 w1,x w2 u2 − u1 w2 , = − = A 1 w1 x w1 w2 w12 w12     w2,y w1 w1,y w2 u2 u1 w2 w1 y = w1 − w2 = B1 w2 − w2



1

1

w2 yields the representation of case i). On the other the assignment r = w 1 u1 is rational and u2 is not or vice versa, (2.50) has exactly one hand, if w w1 1 rational solution. If neither are rational, there is none. If both are rational u is not, there are exactly two rational solutions. These cases but the ratio w cover alternative ii). Based on this theorem, an algorithm for determining rational solutions of systems (2.50) may be designed as follows. The solution procedure is initiated with the first equation. If its general solution is rational, it contains a

88 rx + p with r, p ∈ Q(x, y). r + C(y) Substituting this expression into the second equation yields the first order equation constant w.r.t. x and has the general form

rx Cy − (py + B1 p2 + B2 p + B3 )C 2   − rxy + 2py r + 2B1 p(rx + pr) + B2 (rx + 2pr) + 2B3 r C −(rxy + py r)r + rx ry − B1 (rx + pr)2 − B2 (rx + pr)r − B3 r2 = 0 (2.55) for the y−dependence of C. Because the dependence on x is completely explicit, it may be decomposed w.r.t. powers of x. The coefficient of C 2 has the form of the second equation of (2.50) for p. Therefore, depending on whether or not p satisfies this equation, this leads to a system of linear or Riccati ode’s. If its general solution is rational, the same is true for the full system. Otherwise special solutions may exist leading to special rational solutions of the full system. If the first equation does not allow a general rational solution, special rational solutions may exist. Those also fullfilling the second equation are special solutions of the full system. The following algorithm is organized according to these steps. Algorithm 2.13 RationalSolutionPartialRiccati1(e1 , e2 ). Given two partial Riccati-like equations (2.50), its general rational solution, or the maximal number of special rational solutions are returned, or failed if none exists. S1 : Rational solutions of first equation. Determine the rational solutions of e1 . If there are none return failed. S2 : Special rational solutions. If the rational solutions obtained in S1 are special, substitute them into e2 an return those that satisfy it. S3 : Solve equation for C. Substitute p and r as obtained in S1 into (2.55) and solve the resulting system of ode’s for C. If it does not have any rational solutions return f ailed, otherwise substitute them into z and return the result. Example 2.65 Consider the equations 2 x 2  2 1 2xz y 2 − = 0, z + z + − z = 0. zx + z + y 2 2
2 2 y y x2 − x2 − y− 2 y y x Its coefficients obey the coherence conditions (2.51). In step S1 of the above algorithm the general solution of the first equation is obtained in the form
1 xy ¯ z x, y, C(y) = ¯ − . C(y) + x x2 y − 2 In step S3 the single equation C¯y + C¯ = 0 for the y-dependence of C¯ is obtained. Its solution C¯ = C y with C a constant yields the general rational y xy solution z = xy + C − 2 of the full system. x y−2

Linear Differential Equations

89

The system (2.50) is invariant w.r.t. exchange of x and y. Therefore it makes no difference if the algorithm is started by solving the equation determining the y-derivative, this is the subject of Exercise 2.25. The special case B1 = 0 of (2.50), i.e., a Riccati equation for the xdependence and a linear equation for the y-dependence is of some interest in later applications. In Exercise 2.27 it is shown that the solution to any such system may be written as  1 1  A1,x − A2 + C (2.56) z= 2A1 A1 A1 where C ≡ C(x) satisfies the Riccati equation  2 1  A1,x 1  A1,x Cx + C 2 + − A2 − A2 = 0. +A1 A3 − 2 A1 4 A1 x

(2.57)

The terms not containing C are independent of y. This property makes it into a genuine ode problem. A system closely related to the preceding one is zxx + zx2 + A1 zx + A2 = 0, zy + B1 zx + B2 = 0

(2.58)

which occurs in the analysis of several symmetry classes later on in this book. The coherence conditions for its coefficients are B1,xx + 2B2,x − (A1 B1 )x = A1,y , B2,xx + A1 B2,x − A2 B1,x − (A2 B1 )x = A2,y . The first equation (2.58) is a first order Riccati equation for zx . If its general solution is rational, z is obtained from the system zx =

rx + p, C +r

zy = −

B 1 rx − B1 p − B2 C +r

(2.59)

with C ≡ C(y). The integrability condition for its right hand sides leads to the first order ode
rx Cy − (B1 p + B2 )x + py C 2   − 2r((B1 p + B2 )x + py ) + (B1 rx + ry )x C (2.60) +rx ry − (B1 rx + ry )x r + B1 rx2 = 0 for the y-dependence of C. If (B1 p + B2 )x + py = 0 it is a linear equation, otherwise a Riccati equation. Substituting the solution for C into (2.59), the solution for z is obtained by Lemma 2.10. If the first equation (2.58) has only special rational solutions, they are substituted into the second one and z is again obtained by Lemma 2.10. Partial Riccati-Like Systems: Two Functions. The next Riccati-like system to be considered contains two unknown functions z1 and z2 , depending

90 on x and y. Originally it occured in Lie’s investigation of certain second order ode’s. As will be seen below, the decomposition of various types of Janet bases leads to systems of this kind as well. In order to make the algorithms to be designed more generally applicable, the following class of systems will be considered. e1 ≡ z¯1,x + a0 z¯12 + a1 z¯1 + a2 z¯2 + a3 = 0, e2 ≡ z¯1,y + b0 z¯1 z¯2 + b1 z¯1 + b2 z¯2 + b3 = 0, e3 ≡ z¯2,x + c0 z¯1 z¯2 + c1 z¯1 + c2 z¯2 + c3 = 0,

(2.61)

e4 ≡ z¯2,y + d0 z¯22 + d1 z¯1 + d2 z¯2 + d3 = 0. If a0 d0 6= 0 is assumed, this system is coherent if either of the following sets of constraints is satisfied. b0 − d0 = 0, a0 − c0 = 0, d0,x + b2 c0 − c2 d0 = 0, c3,y − d3,x + a3 d1 − b3 c1 − c2 d3 + c3 d2 = 0, c2,y − d2,x + a2 d1 − b2 c1 − b3 c0 + 2c3 d0 = 0, c1,y − d1,x + a1 d1 − b1 c1 − c0 d3 + c1 d2 − c2 d1 = 0, c0,y − b1 c0 + c1 d0 = 0,

(2.62)

a3,y − b3,x − a1 b3 − a2 d3 + a3 b1 + b2 c3 = 0, a2,y − b2,x − a1 b2 + a2 b1 − a2 d2 + a3 d0 + b2 c2 = 0, a1,y − b1,x − a2 d1 + b2 c1 − 2b3 c0 + c3 d0 = 0 or d1 = 0, c1 = 0, c0 = 0, b2 = 0, b0 = 0, a2 = 0, d0,x − c2 d0 = 0, c3,y − d3,x − c2 d3 + c3 d2 = 0, c2,y − d2,x + 2c3 d0 = 0, a3,y − b3,x − a1 b3 + a3 b1 = 0, a1,y − b1,x − 2a0 b3 = 0, a0,y − a0 b1 = 0. (2.63) If the first alternative (2.62) applies, new dependent variables z1 and z2 are introduced by   1 a + c − ∂x (a0 d0 ) , z¯1 = a10 z1 − 3a 1 2 a0 d0 0 (2.64)   ∂ (a d ) z¯2 = 1 z2 − 1 b1 + d2 − y 0 0 . d0 3d0 a0 d0 The system (2.61) corresponding to conditions (2.62) is transformed into the normalized coherent system e1 ≡ z1,x + z12 + A1 z2 + A2 z1 + A3 = 0, e2 ≡ z1,y + z1 z2 + B1 z2 + B2 z1 + B3 = 0, e3 ≡ z2,x + z1 z2 + B1 z2 + B2 z1 + B3 = 0, e4 ≡ z2,y + z22 + D1 z2 + D2 z1 + D3 = 0.

(2.65)

Linear Differential Equations

91

The new coefficients A1 , . . . , D3 are nonlinear expressions of the old ones a0 , . . . , d3 . Due to the coherence they satisfy A1,y − B1,x − A2 B1 + A1 B2 − A1 D1 + A3 + B12 = 0, A2,y − B2,x − A1 D2 + B1 B2 − B3 = 0, A3,y − B3,x − A2 B3 − A1 D3 + A3 B2 + B1 B3 = 0 B1,y − D1,x + A1 D2 − B1 B2 + B3 = 0, B2,y − D2,x + A2 D2 − B22 + B2 D1 − B1 D2 − D3 = 0, B3,y − D3,x + A3 D2 − B2 B3 − B1 D3 + B3 D1 = 0.

(2.66)

Consider the equations

Example 2.66

y3 3x2 y 2 − 3y 4 2x2 y − y 3 3y 2 ¯1 + 4 z¯2 − = 0, z¯1,y + z¯1 z¯2 + = 0, 3 z 6 x x x x5 2 2 2x2 y + y 3 1 z¯ + 3y z¯ + x + 3y = 0. z¯2,x + z¯1 z¯2 − = 0, z¯2,y + z¯22 + x 1 5 2 2 4 x x x

z¯1,x + z¯12 +

Its coefficients satisfy the conditions (2.62), but z¯1,y 6= z¯2,x . Introducing new y2 y variables by z¯1 = z1 − 3 , z¯2 = z2 − 2 yields x x y y2 y3 3 z1 + 4 z2 = 0 z1,y + z1 z2 − 2 z1 − x x x y y2 1z + z2,x + z1 z2 − 2 z1 − 3 z2 = 0, z2,y + z22 + x 1 x x

z1,x + z12 +

y2 z2 = 0, x3 y z2 = 0, x2

i.e., the normal form (2.65). If the second alternative (2.63) applies, new dependent variables z1 and z2 are introduced by 1 1 z¯1 = z1 , z¯2 = z2 . a0 a0 In this case the system (2.61) corresponding to conditions (2.63) is transformed into z1,x + z12 + A1 z1 + A2 = 0, z1,y + B = 0, z2,x + z22 + C = 0, z2,y + z22 + D1 z2 + D2 = 0, i.e., it decouples into two individual systems of the type (2.50) for z1 and z2 and may be treated by the methods of the preceding subsection. The subsequent theorem describes the structure of the various types of rational solutions of system (2.65). Theorem 2.26 If the coherent system (2.65) has a rational solution, one of the following alternatives applies with r, s, p, q ∈ Q(x, y) and py = qx .

92 i) The general solution is rational and contains two constants. It may be written in the form z1 =

C2 ry + sy C2 rx + sx + p, z2 = + q. C1 + C2 r + s C1 + C2 r + s

ii) There is a rational solution containing a single constant, it may be written in the form z1 =

ry rx + p, z2 = + q. C +r C +r

iii) There is a solution as described in the preceding case, and in addition there is a special rational solution not equivalent to it. iv) There is a single one, or there are two or three special rational solutions which are pairwise inequivalent. Proof The coefficient constraints are the integrability conditions for the u , z = v yields system (2.65). Substituting z1 = w 2 w u(u − wx ) + A1 u + A2 v + A3 w = 0, w u(v − wy ) uy + + B1 u + B2 v + B3 w = 0, w v(u − wx ) + B1 u + B2 v + B3 w = 0, vx + w v(v − wy ) vy + + D1 u + D2 v + D3 w = 0. w

ux +

In order to obtain a linear system for u, v and w, the constraints wx − u + P w = 0, wy − v + Qw = 0

(2.67)

are necessary where P, Q ∈ Q(x, y) still have to be determined. Substituting these conditions into the above system yields ux + (A1 + P )u + A2 v + A3 w = 0, uy + (B1 + Q)u + B2 v + B3 w = 0, vx + B1 u + (B2 + P )v + B3 w = 0, vy + D1 u + (D2 + Q)v + D3 w = 0. (2.68) The consistency of the substitutions for z1 and z2 requires that the system comprising the equations (2.68) and (2.67) be a Janet basis. This is assured if there holds Py − Qx + B3 − B4 = Py − Qx + 13 (A1,y − B1,x + B2,y − D2,x ) = 0. The second form has been obtained by applying equations (2.66). These conditions are satisfied if P = − 13 (A1 + B2 ) and Q = − 13 (B1 + D2 ) are

Linear Differential Equations

93

chosen. It yields the linear system ux + ( 32 A1 − 31 B2 )u + A2 v + A3 w = 0, uy + ( 32 B1 − 13 D2 )u + B2 v + B3 w = 0, vx + B1 u − ( 13 A1 − 23 B2 )v + B3 w = 0, vy + D1 u − wx − u −

1 3 (A1

( 13 B1

−

2 3 D2 )v

(2.69)

+ D3 w = 0,

+ B2 )w = 0, wy − v − 31 (B1 + D2 )w = 0

that is equivalent to (2.61). Let its general solution be u = C1 u1 + C2 u2 + C3 u3 , v = C1 v1 + C2 v2 + C3 v3 , w = C1 w1 + C2 w2 + C3 w3 where wk 6= 0 for all k. The corresponding solutions of (2.61) are z1 =

C1 u1 + C2 u2 + C3 u3 C1 v1 + C2 v2 + C3 v3 u v = = , z2 = . w C1 w1 + C2 w2 + C3 w3 w C1 w1 + C2 w2 + C3 w3

uk and vk are rational and the remaining If exactly one or two of the ratios w wk k ones are not, system (2.61) has the corresponding number of rational solutions. u or v are not, there are exactly three If all three ratios are rational but w w rational solutions. These alternatives combined make up case iv). u and v are rational it may occur that they become rational If not both w w if one of the integration constants vanishes, e. g. C3 . This yields u2 v2 w2 w2 − u1 2 − v1 2 w1 w1 C1 u1 + C2 u2 C1 v1 + C2 v2 w1 u1 w1 v1 z1 = = w2 + w , z2 = C w + C w = w2 + w . C1 w1 + C2 w2 1 1 1 2 2 1 C+ C+ w1 w1 Applying the relations w  k

w1

wk,x wk w1,x wk uk − = − u1 2 , 2 w1 w1 w1 w1

=

wk,y wk w1,y vk wk − = − v1 2 w1 w1 w12 w1

x

w  k

w1

=

y

which are a consequence of the last two equations of (2.69), the solutions of u3 and v3 are rational, this case ii) are obtained. If in addition the ratios w w3 3 is case iii). u and v both are rational. These quotients may be Assume now that w w written as u w2  u3 w3 2 u1 u2 u3 C2 − u1 2 + − u1 2 C1 + C2 + w1 w1 u u1 w1 w1 w1 w1 w1 z1 = = = + , w w w w 2 3 2 3 w w1 C1 + C2 + C1 + C2 + w1 w1 w1 w1

94

z2 =

u = w

v w2  v3 w3 2 v1 v2 v3 C2 − v1 2 + − v1 2 + C2 + w w v1 w w 1 1 w1 w1 w1 1 1 + w2 w3 = w2 w3 w1 C1 + C2 + C1 + C2 + w1 w1 w1 w1

C1

where C1 and C2 have been appropriately redefined. Applying again the above k expressions for the derivatives of w w1 , the representation of case i) follow. In order to determine the rational solutions of system (2.65) a more general problem will be considered first. Let u and v be two functions of x and y. Furthermore, let the x-dependence of u be determined by a Riccati ode in x of order n, and the y-dependence of v by a Riccati ode in y of order m, both m, n ≥ 1. In addition uy = vx is required, i.e., the full Riccati-like system is Rxn (u) = 0,

Rym (v) = 0,

uy − vx = 0.

(2.70)

The structure of its rational solutions is described next, it is the analogue of Theorem 2.3 for pde’s in two dependent and two independent variables. Theorem 2.27 Let ri , si ∈ Q(x, y) and Pi = {pi,1 , . . . , pi,k }, pi,j ∈ Q[x, y], ¯ the sets Pi linearly independent over Q, j = 1, . . . , k, i = 1, . . . , m Ci,j ∈ Q, and define SPrii,si

Pk Pk ∂x ( j=1 Ci,j pi,j ) ∂y ( j=1 Ci,j pi,j ) ≡ {ri + Pk , si + Pk }. j=1 Ci,j pi,j j=1 Ci,j pi,j

The rational solutions of (2.70) are the disjoint union of SPrii,si , i = 1, . . . , m for some natural numbers k and m. The proof of this result may be found in the article by Li and Schwarz [104]. The main steps for designing an algorithm that determines the rational solutions as described by the above theorem will be given next. At first the two given Riccati ode’s are solved by applying the algorithm RationalSolutionRiccati on page 21. If rational solutions do exist, it returns expressions of the form given in Theorem 2.3. The Ci,j are constants w.r.t. x or y now. Their dependence on y and x respectively has to be determined such that the structure described in Theorem 2.27 is achieved. The last equation of the system (2.70) requires that its solutions be integrable pairs. In particular this is true for the terms ri and si occuring in SPrii,si because the terms involving the sums over the pi,j are already integrable pairs by themselves. If (ri , si ) is not an integrable pair, but two polynomials have been found that make them into an integrable pair, the obvious relation P P ∂x ( Ci,j hpi,j ) ∂x ( Ci,j pi,j ) ∂x h P = P + (2.71) Ci,j hpi,j Ci,j pi,j h which is valid for any h ∈ Q[x, y] may be applied to move over the logarithmic derivative of h to ri and si without changing the full expression SPrii,si . If for

Linear Differential Equations

95

a given value of i it is not possible to make (ri , si ) into an integrable pair by this procedure, the respective SPrii,si has to be discarded. In order to make the integrable pairs obtained in this way into a genuine solution of the full system (2.65), the y- and the x- dependence of the constants w.r.t. x and y respectively have to be determined in a final step. Due to the fact that the x-dependence is completely explicit in the solution of the Riccati equation in x, a bound may be obtained for the polynomials in x representing the constants in y occuring in the solution of the Riccati equation in y, and vice versa. This leads to a finite linear problem over the constants, again details may be found in [104]. These steps are organized in terms of the subsequent algorithm. Algorithm 2.14 RationalSolutionPartialRiccati(Rxn ,Rym ). Given a partial Riccati-like system of the form (2.70), its rational solutions are returned, or failed if none exists. S1 : Solve Riccati ode’s. Determine the rational solutions of Rxn and Rym by applying the algorithm RationalSolutionRiccati. If none exists neither for Rxn nor for Rym , return failed. S2 : Generate solution candidates. From the solutions obtained in S1 form all possible sets SPrii,si and retain those that can be made into an integrable pair. If none contains a constant, return failed. S3 : Generate solutions. For each integrable pair obtained in step S2, solve the linear problem for determining the integration constants. If a consistent solution exists, construct a solution from it. Return the set of all solutions or failed if none has been found. Example 2.67

Consider the Riccati-like system

2 a + 2 = 0, Rx1 (a) ≡ ax + a2 − 3x 3x2

4 b = 0, Ry1 (b) ≡ by + b2 + 3y

(2.72)

2 supplemented by ay − bx = 0. In step S1, the special rational solutions 3x 1 are obtained for a, and − 1 and 0 for b. They yield the integrable and x 3y

1 2 1 2 1
and 1 , 0
in step S2. Because pairs (a, b) = 3x , − 3y , 3x , 0 , x , − 3y x they do not contain any undetermined elements, they are returned as result. Example 2.68

Let the Riccati-like system

Rx2 (a) ≡ axx + 3ax a + a3 = 0 and Ry1 (b) ≡ by + b2 +

2 b=0 y−1

supplemented by ay − bx = 0 be given. In step S1 the rational solutions a=

2C1 (y)x + C2 (y) C4 (x) 1 and b = − 2 C1 (y)x + C2 (y)x + C3 (y) C4 (x)y + C5 (x) y − 1

96 are obtained. The single pair made up from them is left unchanged in step 1 , are an inteS2 because the parts not depending on the C 0 s, 0 and − y − 1 grable pair. In step S3 the bounds 2 and 1 for the x- and the y-dependencies respectively lead to a linear system over the constants from which the final answer is obtained in the form a = x + y1 + C ,

1 . b = x + y1 + C − y − 1

Rational and Hyperexponential Solutions of Linear PDE’s. In this subsection the equations under consideration are linear and may be of order higher than one. The fundamental constraint now is that their solution space be finite-dimensional. This may easily be decided if a Janet basis is generated first. Therefore it will be assumed from now on that the systems under consideration are already in Janet basis form. The general system comprises m functions z1 > . . . > zm depending on n independent variables x1 > . . . > xn in lex order, with coefficients from the base field Q(x1 , . . . , xn ). Similar as for linear ode’s, the solution algorithm depends crucially on the function field in which the desired solutions are searched for. At first solutions in the base field are considered. Two important simplifications may be performed according to Theorem 2.14. First of all, starting from a system involving m functions, it allows to generate m subsystems for a single function zk , k = 1, . . . , m, each of which has a finite-dimensional solution space as well. Secondly, any such subsystem involving a single unknown function, contains as lowest equation a linear ode in the lowest variable xn . Its rational solutions may be determined by RationalSolutionsLode on page 14. As a result, the single unknown function may be expressed as a rational function of this lowest independent variable with coefficients depending on the higher variables. Upon substitution into the remaining equations of the system, a linear system for these coefficients is obtained. It is of the same type as the original system with the number of independent variables diminished by one. Therefore, repeating this procedure n times, all dependencies for the coefficients are removed. In a final step a linear algebra problem has to be solved in order to make the solutions for the m subsystems into a genuine solution of the originally given system for all m functions. The following algorithm is based on this scheme. Algorithm 2.15 RationalSolutionLinearSystem(L). Given a linear homogeneous system L for m functions z1 , . . . , zm , depending on x1 , . . . , xn with coefficients from Q(x1 , . . . , xn ), with finite-dimensional solution space. Its rational solutions are returned, or failed if none exists. S1 : Systems for a single function. Generate a system for each individual function z1 , . . . , zm by taking it in turn as lowest dependent variable in lex order.

Linear Differential Equations

97

S2 : Solve linear ode. For each system generated in S1, determine the general rational solution of the linear ode for the lowest independent variable, and store it on an auxiliary list. If all of them are trivial, return failed. S3 : Generate systems for coefficients. Substitute the rational solutions obtained in step S2 into the remaining equations of the respective system. If the coefficients still carry a dependency with them, go to S1. S4 : Solve linear algebraic system. Substitute the relations obtained in S2 and S3 into the original system L. Solve the resulting algebraic system, construct a solution from it and return the result. The following example applies this algorithm, it originates from the symmetry analysis of equation 6.91 from Kamke’s collection. Consider the system

Example 2.69

2y 2 z = 0, z − 2 z − 1 z = 0, z1 + 3x 2 2,y 3x 1 3y 2 3x2 4 z + 1 z = 0, z − 1 z + x z = 0. z1,x − 3x 1 1,y 3y 2 3y 1 3y 2 2

z2,x −

(2.73)

In step S1 the two systems for z1 and z2 are generated. z1,xx −

2 x 1 2 z1,x + 2 z1 = 0, z1,y − z1,x + z1 = 0. 3x y y 3x

(2.74)

4 x 1 z2,x = 0, z2,y − z2,x − z1 = 0. 3x y y

(2.75)

z2,xx +

In step S2, the lowest equations for z1 and z2 yield the rational solutions z1 = C1 (y)x and z2 = C2 (y). Substituting them into the remaining equation of the respective system in S3 yields C1,y = 0 and C2,y − y1 C2 = 0. Due to their dependencies, the algorithm returns to S1. There is nothing to do here because only a single equation is left in either case. Step S2 generates the solutions C1 (y) = C3 and C2 = C4 y with C3 , C4 constants. There is no action in step S3. The linear algebraic system in step S4 reduces to the single equation C3 − C4 = 0. So the final answer is z1 = Cx, z2 = Cy, it generates a one-dimensional solution space. Because the general solution of (2.73) allows a two-dimensional solution space, this result shows that the general solution is not rational, i.e., there must be an additional basis element which is not rational, it will be determined in the next example. The last problem to be considered in this chapter is to find hyperexponential solutions of linear homogeneous systems for z1 , . . . , zm . Here the number of independent variables is restricted to two, usually x and y. This is sufficient for the applications in the symmetry analysis in later chapters. The discussion follows closely the article by Li and Schwarz [104]. At first m = 1 is assumed, i.e., a system for a single function z depending on x and y with coefficients from

98 Q(x, y) is considered. By Theorem 2.14 two linear ode’s w.r.t. x and y may be generated. For both of them the associated Riccati equation for a function z u ≡ zzx and v ≡ zy is constructed as explained on page 16. These equations are supplemented by the obvious integrability condition uy = vx which makes u and v into an integrable pair. This subsystem is of type (2.70), its rational solutions may be determined by the algorithm RationalSolutionPartialRiccati. They correspond to hyperexponential solutions of the original linear system. The remaining equations of the system for z generate linear relations between the constants such that the solution space of the full system is generated. The following algorithm applies these steps. Algorithm 2.16 HyperexponentialSolutionLinearSystem(L). Given a linear homogeneous system L for a function z, depending on x and y, with a finite-dimensional solution space, its general hyperexponential solution is returned, or failed if none exists. S1 : Generate Riccati-like system. Generate two Janet bases in lex order from L with x > y and y > x. From the two lowest equations generate associated Riccati equations in x and y respectivey. S2 : Solve Riccati-like system. Determine the rational solutions of the Riccati-like system corresponding to the equations obtained in step S1 by algorithm RationalSolutionPartialRiccati. If none is found, return failed. S3 : Generate systems for coefficients. From the solutions obtained in step S2 construct the hyperexponential solutions, substitute them into the remaining equations of the linear system and generate the linear system for the coefficients. S4 : Solve linear algebraic system. Solve the linear algebraic system obtained in step S3, construct a solution from it and return the result. If in a system m functions z1 , . . . , zm are involved, at first m subsystems for each variable zk , k = 1, . . . , m are generated. To each of them the above algorithm is applied, and in a final step an additional linear system for the coefficients has to be solved in order to generate the solution of the full system. Example 2.70 In order to find the hyperexponential solutions of (2.73), system (2.74) for z1 will be considered first. In step S1, from the two lowest equations 2 4 2 z1,x + 2 z1 = 0, z1,yy + z1,y = 0 (2.76) z1,xx − 3x 3y 3x the Riccati equations (2.72) are generated. Its rational solutions have been determined in Example 2.67. The four integrable pairs obtained there yields the general solution z = C1

x2/3 y

1/3

+ C2 x2/3 + C3

x y

1/3

+ C4 x.

Linear Differential Equations

99

Substitution of this expression into (2.74) leads to the relations C1 = C4 = 0, C2 − C3 = 0; consequently, the full two-dimensional solution space of (2.74) 2/3 is generated by x and x1/3 . y It turns out that system (2.75) leads to two equations for z2 which may be obtained from (2.76) by exchange of x and y, i.e., two basis elements for its y 2/3 solution space are y and 1/3 . Finally, if a general linear combination of these x basis elements is substituted into (2.73), a linear system for the coefficients is obtained, the solution of which yields the final answer z1 = C1 x + C2

x2/3 y

, 1/3

z2 = C1 y + 2C2

y 2/3 x1/3

.

This is the full solution space of system (2.73). Decomposing Janet Bases. The hyperexponential solutions discussed in the preceding section correspond to first order right factors. For Janet bases of order d three or higher, right factors of any order less than d may occur. A complete understanding of the solution space determined by such a Janet basis is only possible if these right factors may be found. To this end the analogue of the Loewy decomposition of linear ode’s is required which in turn requires a generalized Beke-Schlesinger scheme for these pde’s. For the applications later in this book the general case is not needed. Therefore only a few results for a certain third order Janet basis which occurs frequently in the symmetry analysis will be described. They are based to a large extent on a recent article by Li et al. [105]. For a nontrivial exact quotient q of two linear ode’s l1 and l2 to exist, i.e., for a relation l1 = ql2 to be valid, it is necessary that the order of l2 be strictly less than that of l1 . For Janet bases the situation is more complicated. The partial order defined in the subsequent lemma describes the possible candidates for exact division for the Janet basis types J (1,2) and order not higher than three. The following shortened notation is introduced for describing these order relations: a ≺ {b, c, d, . . .} means a ≺ b, a ≺ c, a ≺ d . . . . Lemma 2.13 For Janet bases of type J (1,2) and order not higher than three, the relations (1,2)

J1 (1,2)

J2,1

(1,2)

(1,2)

(1,2)

≺ {J2,1 , J2,2 }, (1,2)

(1,2)

≺ {J3,1 , J3,2 }, J2,2

(1,2)

(1,2)

≺ {J3,2 , J3,3 }

define a partial order that may be represented by the diagram

100 (1,2)

(1,2)

J3,1

(1,2)

J3,2

J3,3

@

@ @ R @

(1,2) J2,1

@ R @



(1,2)

J2,2



@ @ R @

(1,2)

J1



The exact quotient of two Janet bases e and f of these types may exist only if f is lower than e in this partial order. Proof The partial order property is obvious. The leading derivative zy (1,2) in the type J3,1 Janet basis cannot be reduced by any element of a type (1,2)

J2,2

Janet basis, therefore the latter can never divide the former. The same (1,2)

is true for the leading derivative zx of a type J3,3 (1,2) J2,1

Janet basis and a type

Janet basis. Similar arguments apply for the remaining cases. (1,2)

The Janet basis of type J3,2 in the upper line of the above diagram occurs frequently in the symmetry analysis of ode’s. Therefore the complete answer for its decomposition involving all possible Janet bases of lower order is given explicitly. Theorem 2.28

A Janet basis

{zxx + A1 zy + A2 zx + A3 z, zxy + B1 zy + B2 zx + B3 z, zyy + C1 zy + C2 zx + C3 z} (1,2)

of type J3,2

may have the following components. (1,2)

i) A type J1 component {zx + az, zy + bz} iff its coefficients a and b are rational solutions of ax − a2 + A2 a + A1 b − A3 = 0, ay − ab + B2 a + B1 b − B3 = 0, bx − ab + B1 b + B2 a − B3 = 0, by − b2 + C1 b + C2 a − C3 = 0. (2.77) (1,2)

ii) A type J2,1

component {zy + a1 zx + a2 z, zxx + b1 zx + b2 z} with

w1 , a = w2 , b = A w1 + A , b = −A w2 + A , a1 = − w (2.78) 2 1 1 w0 2 2 1 w0 3 w0 0  1 A2 B1 A2  w0,x − w0 , w2 = −w0,y − w0,x − B2 − B1 w0 A1 A1 A1 A1 (2.79) if A1 6= 0 and w0 is a hyperexponential solution of a coherent third order system of type {w0,xx , w0,xy , w0,yy }. w1 = −

Linear Differential Equations

101 (1,2)

iii) A component {zx + a1 z, zyy + b1 zy + b2 z} of type J2,2 if A1 = 0. Then the coefficients are a1 = B1 , b1 = C1 and b2 = C3 − B1 C2 . Proof The coefficient constraints for case i) and case iii) are obtained by direct reduction. The system (2.77) has been discussed in detail on page 90. For case ii) let z1 and z2 be a fundamental system for a possible factor of (1,2) type J2,1 and define z1,x z2,x z1,y z2,y z1,y z2,y . w0 = , w1 = , w2 = z1 z 2 z 1 z2 z1,x z2,x

(2.80)

The expressions (2.78) for the coefficients are obtained by substituting z1 and (1,2) (1,2) z2 into the type J2,1 system, using the given type J3,2 system for reduction and solving the respective linear algebraic systems. By differentiation and (1,2) reduction w.r.t. to the given type J3,2 Janet basis the following first order system for the determinants w0 , w1 and w2 is obtained. w0,x + A1 w1 + A2 w0 = 0, w0,y + w2 + B1 w1 + B2 w0 = 0, w1,x − w2 + B1 w1 + B2 w0 = 0, w1,y + C1 w1 + C2 w0 = 0, w2,x + (A2 + B1 )w2 + A3 w1 − B3 w0 = 0,

(2.81)

w2,y + (B2 + C1 )w2 + B3 w1 − C3 w0 = 0. The coherence follows from the integrability conditions given in Theorem 2.15. The expressions (2.79) for w1 and w2 are obtained from the first two equations. If they are applied for elimination, due to the coherence the resulting (1,2) system for w0 may be transformed into a third order type J3,2 Janet basis in grlex term ordering with y > x. The details are discussed in Exercise 2.12. (1,2) As a result the type J3,2 system for w0 is obtained explicitly for general coefficients A1 , A2 , . . . , C3 . Although the system for w0 obtained above appears quite cumbersome, it is useful for implementing decomposition algorithms because it is obtained without any Janet basis calculation. Example 2.71 has a type of

(1,2) J1

(1,2)

If the type J3,2

Janet basis considered in Example 2.55

component, according to (2.77) its coefficients are solutions

4 a − 2 = 0, a − ab + 1 b = 0, ax − a2 + x y x x2 1 1 x 2 bx − ab + x b = 0, by − b + y b − 2 a + 22 = 0. y y

(2.82)

5 , b = −¯b + 1 in this case The transformation (2.64) which is a = −¯ a + 3x 3y

102 leads to 1a 2 ¯b + 2 = 0, 2 a ¯ − 2 2 = 0, a ¯y + a ¯¯b − 3y ¯ − 3x a ¯x + a ¯2 + 3x 9xy 9x 2 1 2 x 1 2 ¯bx + a ¯ − 3x ¯b + 9xy = 0, ¯by + ¯b − 2 a ¯¯b − 3y a ¯ + 3y ¯b − 2 2 = 0 y 9y

(2.83)

with the general solution C2 y1 + y 1 a ¯= − , 3x C1 + C2 x + xy y

C2 x2 − x 1 y ¯b = − + . 3y C1 + C2 x + xy y

2 , ¯b = 2 ¯ = − 3x The special choice C1 = 0, C2 −→ ∞ yields the solution a 3y 1 , b = 1 . Due to the fact that the general solution of the and finally a = x y (1,2) Riccati-like system for a and b is rational, the originally given type J3,2 Janet basis is completely reducible. It is left as an exercise to determine the remaining components of its Loewy factor.

Exercises Exercise 2.1

In the following system of algebraic equations x3 + a4 x2 + a3 y + a2 x + a1 = 0, xy + b4 x2 + b3 y + b2 x + b1 = 0, y 2 + c4 x2 + c3 y + c2 x + c1 = 0

x and y are considered as indeterminates, the ai as parameters. Find the constraints for the parameters ak , bk and ck such that the left hand sides of the above equations are a Gr¨obner basis in grlex ordering with y > x. Exercise 2.2 Let a third order ode y 000 + py 0 + 21 p0 y = 0 with p ≡ p(x) be given. Show that a fundamental system yi , i = 1, . . . , 3 of the form y1 = z12 , y2 = z1 z2 , y3 = z22 exists where z1,2 is a fundamental system of z 00 + 14 pz = 0. Later on it will be seen that an analogous property holds for the n − th order linear homogeneous equation with suitable constraints for its coefficients and is a consequence of its symmetry structure. Exercise 2.3 Generalize Example 2.2 to a third order lode y 000 + q1 y 00 + q2 y 0 + q3 y = r and apply the result to the special equation y 000 = r. Exercise 2.4 Let a second order linear homogeneous ode be given. If it is completely reducible it may be represented in the form y 00 + P y 0 + Qy = Lclm(D + p, D + q)y = 0 with p 6= q.

Linear Differential Equations

103

Show that P and Q have the representation P =p+q−

p0 − q 0 , p−q

Q = pq −

p0 q − q 0 p . p−q

Give an explicit expression for the general solution of the inhomogeneous equation y 00 + P y 0 + Q = R. Exercise 2.5 Determine the solutions corresponding to the various types of nontrivial Loewy decompositions of second order equations as given after Corollary 2.5. Exercise 2.6

Prove Theorem 2.1 for second order Riccati equations.

Exercise 2.7

Prove Corollary 2.6.

Exercise 2.8

Prove Theorem 2.9 for m = 2. (1,2)

Exercise 2.9 Express the coefficients of the type J2,1 Janet basis {zy + a1 zx + a2 z, zxx + b1 zx + b2 z} in terms of a fundamental system {z1 , z2 }. (1,2)

Exercise 2.10 The same problem for the type J2,2 {zx + a1 z, zyy + b1 zy + b2 z} with fundamental system {z1 , z2 }.

Janet basis

Exercise 2.11 Determine by direct reduction a system of pde’s for the (1,2) (1,2) coefficients of a type J2,1 component for a type J3,2 Janet basis and compare the result with Theorem 2.28. Exercise 2.12 explicitly.

Determine the system for w0 of case ii) in Theorem 2.28

Exercise 2.13 Determine the order of a pole in function (2.19) if q(x) has a pole of order M at x = x0 . Distinguish the cases M > 1 and M = 1. Exercise 2.14

(1,2)

Assume the type J2,1

Janet basis

zy + A1 zx + A2 z = 0, zxx + B1 zx + B2 z = 0 has the first order Loewy factor zx + az = 0, zy + bz = 0. Express the general solution in terms of integrals. Exercise 2.15

(2,2)

The same problem for the type J2,3

Janet basis

z1,x + A1 z2 + A2 z1 = 0, z1,y + B1 z2 + B2 z1 = 0, z2,x + C1 z2 + C2 z1 = 0, z2,y + D1 z2 + D2 z1 = 0 with the first order Loewy factor z2 + az1 = 0, z1,x + bz1 = 0, z1,y + cz1 = 0. Exercise 2.16 Let zx + ai z = 0, zy + bi z = 0, i = 1, 2 be two first order systems. Determine its Lclm. Exercise 2.17 Apply the results of Example 2.47 to obtain a basis for the solution space of the corresponding system of pde’s.

104 Exercise 2.18 Let the system zy + a1 zx + a2 z = v, zxx + b1 zx + b2 z = w of inhomogeneous pde’s for z(x, y) be given where a1 , a2 , b1 , b2 , v, w ∈ Q(x, y) (1,2) and the left hand sides form a type J2,1 Janet basis. Express a special solution in terms of an integral if a fundamental system z1 and z2 of the corresponding homogeneous system is known. Exercise 2.19 The same problem for zx + a1 z = v, zyy + b1 zy + b2 z = w (1,2) where the left hand sides form a type J2,2 Janet basis. Exercise 2.20 The same problem for zy + a1 zx + a2 z = v, zxxx + b1 zxx + (1,2) b2 zx + b3 z = w where the left hand sides form a type J3,1 Janet basis and z1 , z2 and z3 are a fundamental system of the corresponding homogeneous system. Exercise 2.21

Show how (2.46) is obtained from (2.45).

Exercise 2.22 Apply Theorem 2.20 to solve the equation zy + azx = r for z(x, y) where a ≡ a(x, y), r = b(x, y)z + c(x, y). 3 = 0 has been The equation ∂I + A ∂I + 2I ∂A + ∂ A ∂x ∂y ∂y ∂y 3 considered by Drach [38], page 364. Use the result of the preceding exercise to describe its solution.

Exercise 2.23

Exercise 2.24

Prove Theorem 2.24.

Exercise 2.25 Determine the general rational solution of the system considered in Example 2.65 by solving the second equation first. Exercise 2.26 Assume a special solution z1 for system (2.50) is known. Show that a second solution may be obtained by solving a linear system. Exercise 2.27 Prove the representation (2.56) for the rational solutions of zx + A1 z 2 + A2 z + A3 = 0, zy + B2 z + B3 = 0. Exercise 2.28

Prove Lemma 2.11.

Chapter 3 Lie Transformation Groups

This chapter describes the basic facts of Lie’s theory of continuous groups in a form that is suited for the applications to differential equations in later parts of this book. Originally Lie considered this only as an auxiliary subject for his main goal, i.e., solving differential equations in closed form by analogy with Galois’ theory of algebraic equations. After recognizing the fundamental importance of these groups for this latter problem he developed a fairly complete theory for them. Its original objective, solving differential equations, was almost completely forgotten, most textbooks on differential equations do not even mention it at all. Later on the theory of continuous groups became a field of independent interest under the name Lie groups. The same is true for the algebraic objects introduced by Killing that were baptized Lie algebras by Hermann Weyl. They were obtained by abstraction from the commutators of vector fields occurring in Lie’s theory. In the first Section 3.1, basic concepts of the theory of continuous groups are put together for later reference, usually without proof. In Section 3.2 below, some properties of abstract Lie algebras are discussed. Section 3.3 applies the previous results to transformation groups of a two-dimensional manifold because these groups are most important for later parts of the book. Classifications of low-dimensional Lie algebras and transformation groups of a two-dimensional manifold are given in Section 3.4. The subject of Section 3.5 are so called Lie systems. These are special systems of linear homogeneous pde’s with the additional feature that their solutions define the coefficients of infinitesimal generators of Lie transformation groups. These results are applied later in the symmetry analysis of differential equations.

3.1

Lie Groups and Transformation Groups

This section follows closely the article by Gorbatsevich and Onishchik in vol. 20 of the Encyclopaedia of Mathematical Sciences [43], part I; see also Chapter 1 of the book by Olver [140], or Chapters 5, 6 and 7 of Ibragimov’s book [78].

105

106

Lie Groups. The main ingredients for this section are the algebraic concept of a group and the differential-geometric notion of a smooth manifold. The term smooth constrains the overlap functions of any coordinate chart to be C ∞ functions. These concepts are merged by the following definition which is the foundation for the subject of this chapter. Definition 3.1 (Lie group) A set G is called a Lie group if there is given a structure on G satisfying the following three axioms. i) G is a group. ii) G is a smooth manifold. iii) The group operations G × G → G, (x, y) → xy and G → G, x → x−1 are smooth functions. If r is the dimension of the manifold, the group is called an r-parameter Lie group and is denoted by Gr . For the applications in this book usually one is not interested in the full Lie group, but only in that part which is connected to the identity element. Starting with any global Lie group, a local Lie group may be constructed by taking a coordinate chart for the group manifold containing the identity, and retaining only those elements corresponding to parameter values contained in this identity neighborhood. From now on, the term Lie group will always be used in this sense, i.e., meaning the local group just described. Let G and H be Lie groups. A smooth map f : G → H is a homomorphism if it is simultaneously a homomorphism of abstract groups and a smooth map of the group manifolds. A homomorphism is called an isomorphism if there exists a smooth inverse f −1 : H → G. Two Lie groups are called isomorphic if an isomorphism between them exists. Due to the restriction to local groups, it is sufficient that these maps be defined for a neighborhood of the identities. A subset H of a Lie group G is called a Lie subgroup if H is a subgroup of the abstract group G and a submanifold of the manifold G. If H is even a normal subgroup of the abstract group G, it is called a Lie normal subgroup. A Lie group G is called simple if the only normal Lie subgroups are the identity and G itself. The commutator group G(1) = G0 = [G, G] is the group generated by all commutators aba−1 b−1 with a, b ∈ G. The higher commutator groups 0 are defined recursively by G(ν+1) = G(ν) for ν = 1, 2, . . .. A Lie group Gr is called integrable or solvable if its series of commutator groups G(ν) terminates after a finite number of steps with the identity. A Lie group is called semi-simple if it does not contain a solvable normal subgroup. Any Lie group has a maximal solvable normal subroup. Let Gr be a r-parameter Lie group and a, b ∈ Gr be two elements with the representation a = (a1 , . . . , ar ) and b = (b1 , . . . , br ) in a coordinate chart. For the coordinates of the product c = ab there holds cj = φj (a1 , . . . , ar , b1 , . . . , br ) for j = 1, . . . , r

(3.1)

Lie Transformation Groups

107

where φj are smooth functions of all arguments. Assume further now that the coordinate chart is such that the identity is represented by e = (0, . . . , 0). Then the lowest terms of (3.1) are P j (3.2) ak bl + . . . cj = aj + bj + 12 k,l γkl j with constants γkl . The omitted terms are of order three or higher with respect to ak and bl . The numbers j j cjkl = γkl − γlk

(3.3)

are anti-symmetric in k and l, they are called the structure constants of the Lie group. Lie Transformation Groups. Let M be a n-dimensional smooth manifold and G a Lie group. An action T of the group G on M is a smooth mapping T : G × M → M,

T (g, x) ≡ gx → x ¯

(3.4)

with the following properties: T (e, x) = x,

T (a, T (b, x)) = T (ab, x)

(3.5)

for any x ∈ M , g, a, b ∈ G, e ∈ G the unit element. Then G is called a Lie transformation group of the manifold M . For a r-parameter group Gr , in local coordinates the action (3.4) has the special form x ¯i = fi (x1 , . . . , xn , a1 , . . . , ar )

(3.6)

with smooth functions fi , i = 1, . . . , n. In this form Lie originally introduced his continuous groups. Specializations of (3.6) are the basis for the symmetry analysis in later parts of this book. An additional assumption is that the number r of parameters is minimal, i.e., that it is not possible to introduce a smaller number of parameters generating the same set of transformations. If this is true they are called essential parameters. It is proved in Lie [112], vol. I, pages 13-14, that this is guaranteed if there do not exist equations of the form r X k=1

χk (a1 , . . . , ar )

∂fi (x1 , . . . , xn , a1 , . . . , ar ) =0 ∂ak

for i = 1, . . . , n. Furthermore, it is shown there how a constructive criterion may be obtained from these requirements. From now on it is always assumed that group parameters are essential without explicitly mentioning it. The elements g ∈ G leaving a given element x ∈ M invariant form the stabilizer of x. If T is an action of a group G on M , one can define an equivalence relation on M by x ∼ y ⇐⇒ x = gy for some g ∈ G.

108 The equivalence classes are called the orbits of the action. Every point of x ∈ M is contained in a unique orbit. The totality of orbits generate an invariant decomposition or simply decomposition of the manifold. The action of a transformation group G on M is called transitive if for arbitrary x, x ¯ ∈ M there is a group element g ∈ G such that x ¯ = gx, otherwise it is called intransitive. If the action is transitive there is only a single orbit of if for x1 , . . . , xk ∈ M , xi 6= xj for i 6= j, there is a g ∈ G such that x ¯i = gxi , but the same is not true for k + 1, i.e., there are k orbits. A transitive transformation group G of a manifold M is called imprimitive if M is the union of submanifolds that are permuted by the group. These submanifolds are called systems of imprimitivity of G. A group that is not imprimitive is called primitive. Let Gr be a r-parameter Lie group with coordinates a = (a1 , . . . , ar ) and such that e = (0, . . . , 0). Defining the quantities ξk,i ≡

∂fi ∂ak a1 =...=ar =0

(3.7)

for k = 1, . . . , r and i = 1, . . . , n, expanding (3.6) at the identity and omitting terms of order higher than one in the group parameters there follows x ¯i = xi + ξ1,i a1 + . . . + ξr,i ar + . . . .

(3.8)

The vector fields Xk = ξk,1

∂ ∂ + . . . + ξk,n , ∂x1 ∂xn

k = 1, . . . , r,

(3.9)

are called the infinitesimal generators of the Lie transformation group. They are linearly independent over constants and obey the commutation relations [Xk , Xl ] = Xk Xl − Xl Xk =

r X

cjkl Xj .

(3.10)

j=1

Mathematical objects obeying these relations have been baptized a Lie algebra. They occur in various branches of mathematics and form an independent subfield of algebra. Some of these algebraic properties are discussed in the next Section 3.2. If X = [Xk , Xl ] is the commutator of the two generators Xk and Xl , its coefficients may be expressed as ξi =

n  X j=1

ξk,i

∂ξk,i  ∂ξl,i − ξl,i for i = 1, . . . , n. ∂xj ∂xj

(3.11)

As usual, if q generators X1 , X2 , . . . , Xq do not obey a relation λ1 X1 + . . . + λq Xq ≡ 0

(3.12)

Lie Transformation Groups

109

with λ1 , . . . , λq constants, not all zero, they are said to be linearly independent over constants or simply independent. A generator Xk that may be expressed in the form Xk = λ1 X1 + . . . + λq Xq is said to be dependent on X1 , . . . , Xq . If q generators X1 , X2 , . . . , Xq do not obey a relation φ1 (x1 , . . . , xn )X1 + . . . + φq (x1 , . . . , xn )Xq ≡ 0

(3.13)

with not all φi (x1 , . . . , xn ) identically zero they are said to be unconnected. Any generator Xk that may be expressed in the form Xk = φ1 (x1 , . . . , xn )X1 + . . . + φq (x1 , . . . , xn )Xq is said to be connected with X1 , . . . , Xq . Unconnected generators are obviously independent, but independent generators are not necessarily unconnected. For a transformation group of a n-dimensional manifold there cannot be more than n unconnected generators, though there may be any number of independent generators. If a set of unconnected generators is such that the commutator of any pair is connected with them, they are said to form a complete system. The following rules for manipulating expressions involving a commutator follow immediately from the definition. Let φ(x1 , . . . , xn ) and ψ(x1 , . . . , xn ) be two functions of x1 , . . . , xn , and X, X1 and X2 arbitrary vector fields. There holds X(φ + ψ) = Xφ + Xψ, [φX1 , X2 ] = φ[X1 , X2 ] − X2 φX1 ,

X(φψ) = ψXφ + φXψ, [X1 , φX2 ] = φ[X1 , X2 ] + X1 φX2 .

Let X1 , . . . , Xr be the infinitesimal generators of an r-parameter Lie transformation group G with group parameters a1 , . . . , ar . The vector fields Zi = bi1 (a1 , . . . , ar )

∂ ∂ + . . . + bir (a1 , . . . , ar ) ∂e1 ∂er

Pr where bik = j=1 cjik ej for i = 1, . . . , r and k = 1, . . . , r are the generators of a linear homogeneous transformation group of an r-dimensional manifold with coordinates (e1 , . . . , er ). It is called the adjoint group of G. Lie’s Fundamental Theorems. Lie himself summarized several important results of his theory in terms of three fundamental theorems that are given next. Theorem 3.1 (Lie’s First Theorem) If the equations (3.6) define an r-parameter group, then the x ¯i , considered as functions of x1 , . . . , xn and a1 , . . . , ar , satisfy a system of pde’s of the special form r

X ∂x ¯i = ψj,k (a1 , . . . , ar )ξj,i (¯ x1 , . . . , x ¯n ) ∂ak j=1

110 for k = 1, . . . , r. The determinant of the ψj,k does not vanish identically. On the other hand, if functions fi (x1 , . . . , xn , a1 , . . . , ar ) satisfy a system of differential equations of this form, they define a transformation group. The proof of this theorem may be found in Lie [112], vol. I, Kapitel 2, pages 27-34 and vol. III, Kapitel 25, pages 545-564. The importance of the infinitesimal generators of a group arises from the fact that a group, due to the restriction to local groups as explained above, is uniquely determined by them. This relation is established in the next theorem. Theorem 3.2 (Lie’s Second Theorem) Any r-parameter Lie group (3.6) defines r infinitesimal generators (3.9) satisfying r(r −1) relations of the form (3.10). Conversely, r independent generators (3.9) satisfying the relations (3.10) define a Lie group. The proof of this theorem may be found in Lie [112], vol. III, Kapitel 25, pages 575-590. It is based on the system d¯ xi = λ1 ξ1,i (¯ x1 , . . . , x ¯n ) + . . . + λr ξr,i (¯ x1 , . . . , x ¯n ) dt for the general one-parameter group contained in the r-parameter group (3.6); the λ1 , . . . , λr are constants. Together with the initial conditions x ¯i = xi for t = 0, i = 1, . . . , n, they determine the one-parameter groups uniquely. If for any fixed k, 1 ≤ j ≤ r, λk = ak , λj = 0 for j 6= k are chosen, the system d¯ xi x1 , . . . , x ¯n ) = ξk,i (¯ dak

(3.14)

for the one-parameter group corresponding to the generator Xk is obtained. Whenever the phrase “the group of the Xk ” for a system of infinitesimal generators is used, it is based on Lie’s second theorem and the restriction to the local group determined by them. Theorem 3.3 (Lie’s Third Theorem) If a system of constants cjkl satisfies the relations cjkl + cjlk = 0 and r X

(cjks csli + cjls csik + cjis cskl ) = 0

s=1

for i, j, k, l = 1, . . . , r, there exists a Lie group with these structure constants. The proof of this theorem may be found in Lie [112], vol. III, pages 597 ff. Similarity of Transformation Groups. If any transformation group is given by relations (3.6), an apparently different group may be obtained by introducing new coordinates of both the underlying manifold M and the group manifold G. Two groups related to each other in this way are not essentially

Lie Transformation Groups

111

different. A special term for this relation has been introduced by Lie [112], vol. I, page 24. Definition 3.2 (Similarity of transformation groups) Let two r-parameter transformation groups be given by x ¯i = fi (x1 , . . . , xn , a1 , . . . , ar ) and y¯i = gi (y1 , . . . , yn , b1 , . . . , br ). If there exist diffeomorphisms xi = ϕi (y1 , . . . , yn ) for i = 1, . . . , n and ak = ψ(b1 , . . . , br ) for k = 1, . . . , r such that the two definitions become identical, the groups are called similar. The consequences of a simultaneous change of both coordinates are discussed in detail by Lie [112], vol. I, Kapitel 2 and by Eisenhart [41], §7. The transformation of the infinitesimal generators under a coordinate change of M occurs frequently in later applications. Let the two coordinates (x1 , . . . , xn ) and (y1 , . . . , yn ) be related by xk = φk (y1 , . . . , yn ) and yk = ψk (x1 , . . . , xn ). for i = 1, . . . , n, the transDefining ηi = Xk ψi (x1 , . . . , xn ) x1 =φ1 ,...,xn =φn

formed generator of (3.9) is Yk = η1 (y1 , . . . , yn )

∂ ∂ + ηn (y1 , . . . , yn ) . ∂y1 ∂yn

Of particular importance is the question of how the commutator of two vector fields is transformed under a change of coordinates. The answer is given in the subsequent lemma due to Lie [112], part I, page 84. Lemma 3.1 If two vector fields Xk and Xj in coordinates (x1 , . . . , xn ) are transformed into Y1 and Y2 in coordinates (y1 , . . . , yn ), its commutator [Xk , Xj ] is transformed into [Yk , Yj ]. As a consequence of this lemma the structure constants of a Lie algebra of vector fields are invariant under a coordinate change of the transformed manifold M . This is important because it decouples the algebraic properties of Lie algebras generated by vector fields from its appearance depending on the actual coordinates of M . Given any two transformation groups the problem arises how to decide similarity and, in case the answer is affirmative, how to actually determine the transformation functions between them. The following theorem due to Lie essentially provides the answer. Theorem 3.4 (Lie 1888). In order that two groups with infinitesimal generators X1 , . . . , Xr in coordinates (x1 , . . . , xn ) and Y1 , . . . , Yr in coordinates (y1 , . . . , yn ) respectively be similar, the following conditions are necessary and sufficient. 1. The two groups Pr must have the same structure, i.e., if the relations [Xi , Xj ] = k=1 ckij Xk are valid it must be possible to choose suitable

112 Pr linear combinations Y¯i = j=1 γij Yj with constant γ’s such that the Y¯i Pr satisfy relations [Y¯i , Y¯j ] = k=1 ckij Y¯k with the same structure constants as for the Xi . Pk 2. If X1 , . . . , Xk are unconnected whereas Xk+i = j=1 φi,j (x1 , . . . , xn )Xj for k ≤ r and i = 1, . . . , r − k, the correspondingP generators Y¯1 , . . . , Y¯k k ¯ are also unconnected whereas the relations Yk+i = j=1 ψi,j (y1 , . . . , yn )Y¯j are valid such that the equations φi,j (x1 , . . . , xn ) = ψi,j (y1 , . . . , yn ) are consistent. In particular they should not generate a relation among x1 , . . . , xn on the one hand or y1 , . . . , yn on the other. The proof may be found in Lie [112], part I, pages 353-355. It shows that the question of similarity of two groups may be divided into two independent parts. This separation is based on Lemma 3.1. In the first place, in order to be similar the infinitesimal generators must generate isomorphic Lie algebras. This is a purely algebraic problem that will be discussed in detail in the Section 3.2. Only if both Lie algebras are isomorphic and a basis transformation has been found such that their structure constants are identical, the second part dealing exclusively with the coordinate change of the transformed manifold applies. A special kind of similarity is involved in defining the so-called canonical variables for a one-parameter group. Let the group equations in coordinates (x1 , . . . , xn ) be x ¯i = fi (x1 , . . . , xn , a) with the single group parameter a. There exist always new coordinates (u1 , . . . , un ) such that the group equations are u ¯i = ui for i = 1, . . . , n − 1 and u ¯n = un + b, i.e., any one-parameter group is similar to a group of translations in a single coordinate. In these latter variables the infinitesimal generator is ∂ . In Section 3.3 finding a ∂un transformation to canonical form for groups acting in the plane will be discussed. Furthermore, canonical forms for groups involving more than a single parameter will be discussed. Defining Equations of a Transformation Group. Starting with the infinitesimal generators of a group, a third representation in terms of the defining equations, as Lie [112], vol. I, page 185 called them, may be obtained. By definition, these equations are such that their general solution generates the vector space of the coefficients of the given generators. This representation is particularly important if the groups to be considered occur as symmetry groups of differential equations because they are usually defined in terms of systems of pde’s. Let the infinitesimal generators Xk of a group be given in the form (3.9). The general generator may be written as r

ξ1

X  ∂ ∂ ∂  ∂ + . . . + ξn = + . . . + ξk,n ek ξk,1 ∂x1 ∂xn ∂x1 ∂xn k=1

(3.15)

Lie Transformation Groups

113

Pr with constants ek . This defines the ξi by means of ξi = k=1 ek ξk,i (x1 , . . . , xn ). The ξk,i are given functions of x1 , . . . , xn . It has been shown by Lie [112], vol. I, § 48, page 181 that for the functions ξi a system of linear pde’s always may be constructed such that they form a fundamental system. Due to its importance a special name has been given to it. Definition 3.3 (Defining equations of a group). The coefficients ξ1 , . . . , ξn of the general infinitesimal generator of a Lie transformation group defined by (3.15) satisfy a system of linear homogeneous pde’s. They are called the defining equations of the group. Altogether there are three means now to describe a Lie transformation group. The finite equations (3.6), the infinitesimal generators (3.9) and the defining equations just described. Due to the restriction to local Lie groups they are essentially equivalent. The infinitesimal generators are obtained from the finite equations by differentiation. The converse requires solving linear pde’s by Lie’s second theorem. The same is true for determining the infinitesimal generators if the defining equations are given. The relation between these representations of a group and how to turn from one to another is illustrated by the following diagram. The operations corresponding to the downward arrows at the left can always be performed algorithmically. Contrary to that, there is no generally valid algorithm for the operations at the right corresponding to the upward arrows. Group Equations 6 Differentiation

Solving 1st Order Ode Systems

? Infinitesimal Generators 6 Solving Linear Solving Linear Pde’s Algebraic Equations ? Defining Equations Invariants and Differential Invariants. A function F (x1 , . . . , xn ) is called an invariant of the r-parameter Lie transformation group Gr defined by (3.6) if there holds F (¯ x1 , . . . , x ¯n ) = F (x1 , . . . , xn ). Invariants of this kind are also called absolute invariants in order to distinguish them from another kind of invariants to be defined in Chapter 4. A criterion for invariance may be given in terms of the infinitesimal generators of the group defined by (3.9). A function F (x1 , . . . , xn ) is an invariant of the group if and only if there holds Xk F (x1 , . . . , xn ) = 0 for k = 1, . . . , r. If F is an invariant of a group, then also Φ(F ) with Φ an undetermined function

114 is an invariant. If a group allows more than a single invariant, only the functionally independent invariants are relevant. If the generic rank q for the group generated by X1 , . . . , Xr is defined by   ξ1,1 , ξ1,2 ξ1,n       ξ2,1 , ξ2,2 ξ2,n   rank . . ≡q .. ..         ξr,1 , ξr,2 ξr,n there are n − q independent invariants. If q = n there are no invariants at all and the group is transitive. The proof may be found in Lie [112], vol. I, Kapitel 13 or in the book by Eisenhart [41], §21. The concept of a differential invariant emerges from the fact that a group acts on a manifold the coordinate variables of which may carry some dependencies with them. Whenever this occurs, the vector fields generating the group may be prolonged up to a predetermined order of the respective derivatives. These prolonged generators of the original group are considered as vector fields acting on a higher-dimensional manifold to which the above results may be applied. If different dependencies are assumed, there are different prolongations and consequently diverse types of differential invariants. In the remaining part of this subsection summation over repeated indices is always understood. Definition 3.4 (Prolongation of a vector field) Let the vector field U = ξi (x, u)∂xi +η α (x, u)∂uα be the infinitesimal generator of a finite transformation group on a manifold with coordinates x = (x1 , . . . , xn ) and u = (u1 , . . . , um ) with dependencies uα ≡ uα (x1 , . . . , xn ) for α = 1, . . . , m. ∂ i1 +...+in uα . By definition the Partial derivatives are denoted by uα i1 ,...,in ≡ ∂xi11 . . . xinn k-th prolongation of U is U (k) = U + ζiα

∂ ∂ + . . . + ζiα1 ,...,ik α . α ∂ui ui1 ,...,ik

The functions ζiα1 ,...,ik are recursively defined by ζiα = Di (η α ) − uα s Di (ξs ) and ζiα1 ,...,ik = Dik (ζiα1 ,...,ik−1 ) − uα i1 ,...,ik−1 s Dik (ξs ). Di is the operator of total differentiation w.r.t. xi . Di =

∂ ∂ ∂ ∂ + uα ... + uα + uα i k,i k,l,i α α ∂xi ∂u ∂uk ∂uα k,l

If the vector fields generating a group are prolonged there arises the question as to what the algebraic relations between the prolonged vector fields are. The

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115

subsequent theorem provides the simple answer that the structure constants for the prolonged generators remain unchanged. Theorem 3.5 Let X1 , . . . , Xr be vector fields generating Pr a group acting on a manifold of arbitrary dimension obeying [Xi , Xj ] = k=1 ckij Xk . Furthermore, let there be some sort of dependencies declared between these variables. (l) The prolongations Xi of any order obey the same commutation relations as Pr (l) (l) (l) the X 0 s, i.e., there holds [Xi , Xj ] = k=1 ckij Xk . The proof may be found in Lie [112], vol. I, §130 or Eisenhart [41], §27. The next theorem due to Lie [114], page 760, assures the existence of an infinite number of differential invariants. Furthermore, it shows that only a finite number of them has to be determined by integrating a complete system of pde’s. Theorem 3.6 (Lie 1893) For any continuous group, acting on a manifold in any number of variables carrying certain dependencies, there exists an infinite number of differential invariants. They may be obtained by differentiation from a finite subset of them, it is called a full system of invariants. The first part of this result is obvious. By differentiating a sufficient number of times the number of variables increases until the condition for the existence of invariants mentioned before is obeyed.

3.2

Algebraic Properties of Vector Fields

While Lie developed his theory of continuous groups and of integrating differential equations in Leipzig, the high school teacher Wilhelm Killing, apparently not knowing Lie’s work for a long time, developed a theory of certain algebraic structures that turned out to comprise the commutators of the infinitesimal generators occurring in Lie’s theory as special cases. On Lie’s suggestion, Elie Cartan in Paris took up the subject in his thesis, correcting and completing Killing’s results, and continued for many years working in this field to become one of its most important contributors. Those topics of the resulting theory that will be applied later on are summarized in this section. In addition to Lie’s publications, good references are the thesis of Cartan [25] and the book by Jacobson [82]. For algorithmic questions the article by Rand [151] or the thesis by de Graaf [37] is recommended. A comprehensive treatment including a survey of the more recent literature may be found in the three volumes of the Springer Encyclopaedia [43], entitled Lie Groups and Lie Algebras. A lot of interesting background information is covered in parts II and III of the book by Hawkins [68].

116 General Properties of Lie Algebras. By abstraction from its geometric origin, the algebraic objects obeying commutation relations (3.10) are studied as a subject of interest of its own right, they are called Lie algebras. First of all they have to be defined. Definition 3.5 (Lie algebra). A Lie algebra L is a vector space over a field F equipped with a binary operation, called commutator, satisfying the following constraints for any U, V, W ∈ L and α, β ∈ F . i) [U, V ] = −[V, U ] (anti-commutativity) ii) [αU + βV, W ] = α[U, W ] + β[V, W ] (bilinearity) iii) [U, [V, W ]] + [V, [W, U ]] + [W, [U, V ]] = 0 (Jacobi identity) In this book only finite dimensional Lie algebras are considered; and it is always assumed that the field F has characteristic zero. If not stated otherwise explicitly it will be the complex number field C. A Lie algebra is usually defined in terms of the commutators of basis elements in the form (3.10). The numerical values of the structure constants ckij depend on the basis of the Lie algebra. If two Lie algebras are such that by a suitable change of the basis their structure constants become identical they are considered to be not essentialy different, clearly they are isomorphic. As usual in algebra, this notion of isomorphism establishes an equivalence relation between Lie algebras. As a consequence, a classification of Lie algebras may be performed by associating them to a canonical form which is given by a special basis. For algebras of low dimension canonical forms are given in Section 3.4 on page 135 ff. Lie [112] has shown that deciding isomorphy of Lie algebras is a purely algebraic problem. To this end a suitable basis transformation has to be found such that in the new basis the commutators coincide with a canonical P one. Let ¯1 , . . . , U ¯r with [U ¯i , U ¯j ] = c¯k U ¯ a basis of an r-dimensional Lie algebra be U ij k . The desired basis elements U1 , . . . , Ur with commutation relations Pcanonical P ¯j [Ui , Uj ] = ckij Uk are expressed in terms of the given ones as Ui = αij U with undetermined coefficients αij ∈ F . Substituting these expressions into the canonical commutation relations leads to the system of algebraic equations r X l,m=1

c¯nlm αil αjm =

r X

ckij αkn ,

1 ≤ i < j ≤ r, 1 ≤ n ≤ r

(3.16)

k=1

with the constraint det{αij } 6= 0. This system has been given already by Lie [112], vol. I, page 290, equations (3). Its solution yields the desired transformation. In general it is not unique. It has been discussed in detail by Gerdt [52] applying Gr¨ obner basis methods for determining its solution. Example 3.1 The vector fields considered in Example 3.3 below will be shown to generate a Lie algebra of type l3,2 . In order to determine the transformation to canonical generators explicitly, the new basis elements are written as Ui = αi1 ∂x + αi2 ∂y + αi3 (y∂x − x∂y ). The commutation relations of l3,2

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117

require that the coefficients satisfy α12 α23 − α13 α22 = 0, α11 α23 − α13 α21 = 0, α11 α33 − α13 α31 + α12 = 0, α12 α33 − α13 α32 − α11 = 0, α21 α33 − α23 α31 − α22 = 0, α22 α33 − α23 α32 + α21 = 0 and α13 = α23 = 0. Its general solution is α13 = α23 = 0, α33 = i, α12 = −iα11 and α22 = iα21 . The coefficients α11 , α21 , α31 and α32 remain undetermined. Choosing α11 = α21 = 1, α31 = α32 = 0 the final answer U1 = ∂x − i∂y , U2 = ∂x + i∂y and U3 = i(y∂x − x∂y ) is obtained. Although a priori any Lie algebra with equal dimension is a candidate for the canonical form of any given Lie algebra, it is more efficient to narrow down the possible types in advance by calculating several invariants. This point has been emphasized by de Graaf [37], Section 7.1. The following example is taken from the above mentioned publication by Gerdt [52]. The characteristic polynomial which occurs there is explained below, see eq. (3.17). Example 3.2 Consider the Lie algebra of vector fields v1 = ∂x , v2 = ∂t , v3 = t∂x +∂u , v4 = x∂x +3t∂t −2u∂u with nonvanishing commutators [v1 , v4 ] = [v2 , v3 ] = v1 , [v2 , v4 ] = 3v2 and [v3 , v4 ] = −2v3 . According to definition (3.17) its characteristic polynomial ω(ω − E4 )(ω − 3E4 )(ω + 2E4 ) is squarefree and decomposes into linear factors. Consequently, the Lie algebra l4,5 defined on page 136 is identified as the proper type. A sub-vectorspace M ⊂ L is a subalgebra if the commutator of any two elements from M is again in M. This is written [M, M] ⊂ M. An invariant subalgebra or ideal is such that any commutator with elements from L is in M, i.e., [M, L] ⊂ M. The normalizer of M is the set of all elements l ∈ L such that [l, M] ∈ M. The normalizer of M is a subalgebra. If S is a subset of L the centralizer of S is the set of all elements c ∈ L such that [c, s] = 0 for all s ∈ S. The centralizer of L is called the center. Fundamental problems are to classify all types of composition leading to nonisomorphic Lie algebras, and to determine simpler substructures within a given Lie algebra, i.e., to determine its invariant subalgebras. The characteristic polynomial P ∆(ω) ≡ | Ei ckij − δjk ω| = (3.17) ω r − ψ1 (E)ω r−1 + ψ2 (E)ω r−2 + . . . + (−1)r−1 ψr−1 (E)ω or correspondingly the characteristic equation ∆(ω) = 0 where E1 , . . . , Er are r indeterminates turns out to be an important tool for dealing with these problems. It has been introduced by Killing [89] in order to answer the question in which two-dimensional subalgebras a given element is contained. The number of independent coefficients of the characteristic polynomial is called the rank of the Lie algebra. Important properties of the characteristic polynomial are given next.

118 Theorem 3.7

Let ∆(ω) and the ψk (E1 , . . . , Er ) be defined by (3.17).

i) The coefficient ψk (E1 , . . . , Er ) is a homogeneous polynomial of degree k in E1 , . . . , Er . ¯i = P αi,j Uj , the ψk are ii) If new basis elements are introduced by U ¯1 , . . . , E ¯r ) = ψk (P αi,1 Ei , . . . , P αi,r Ei ). changed according to ψ¯k (E iii) The coefficients ψk (E1 , . . . , Er ) satisfy the differential equations r X σ,τ =1

cτρσ Eσ

∂ψ = 0, ∂Eτ

i.e. they are invariants of the adjoint group. iv) The rank of a Lie algebra is not higher than the multiplicity of the root ω = 0 of the characteristic equation. The proof may be found in Cartan’s Thesis [25], Chapitre II. Property ii) shows that the structure of the coefficients ψk is invariant under a basis change of the corresponding Lie algebra. This property makes the characteristic polynomial into an important tool for determining the type of a Lie algebra as it is shown in the following example. Example 3.3 The Lie algebra generated by the vector fields ∂x , ∂y and y∂x − x∂y has the nonvanishing structure constants c132 = −1 and c233 = 1. They yield the characteristic polynomial (ω 2 + E32 )ω. By Theorem 3.7 the Lie algebra is of type l3,2 with c = −1 in the classification given in Section 3.4. This follows from the fact that it is the only algebra which has a third order and a first order term in its characteristic polynomial, and the dimension of its derived algebra (see below) has dimension 2. This example is of particular importance because it applies exactly the kind of reasoning that is used in Chapter 5 for identifying the symmetry types of differential equations. Solvable Lie Algebras. The so-called integrable or solvable groups and the corresponding Lie algebras play an important role in the symmetry analysis of differential equations. This term is closely connected with their application for solving linear pde’s as described in Theorem 2.23. The special structure of the commutators of the vector fields admitted by the given linear pde in this theorem is generalized as follows. The derived Lie algebra L0 ≡ L(1) is generated by all commutators of L, i.e., L0 = [L, L]. The derived series is the (k) sequence L, L0 , . . . , L where L(j) = [L(j−1) , L(j−1) ]. Definition 3.6 (Solvable Lie algebra) A Lie algebra of dimension r is called integrable or solvable if L(k) = 0 for some positive integer k ≤ r.

Lie Transformation Groups

119

The following theorem due to Lie and Engel provides a convenient means for identifying solvable Lie algebras if they are given by basis vectors and structure constants. Its proof may be found in Cartan’s Thesis [25], Th´eor`eme IV on page 45. Theorem 3.8 Let L be an r-dimensional Lie algebra with structure constants ckij . Either of the following two conditions is necessary and sufficient for L to be solvable. i) The derived algebra of L has rankPzero. r ii) For i, j, k = 1, . . . , r there holds λ,µ,ν=1 (cµiλ cνjµ cλkν − cλiµ cµjν cνkλ ) = 0. Example 3.4 Consider the three- and four-dimensional algebras listed on page 135. The first and the second derived algebras of l3,1 and l4,1 have dimension 3, therefore by definition they are not solvable. The second or the third derived algebras of the remaining algebras l3,2 to l3,6 and l4,2 to l4,17 have zero dimension, consequently they are solvable. Important properties of solvable algebras are described next, they may be found in Cartan’s Thesis [25] on pages 45-48. Theorem 3.9 ties.

Solvable Lie algebras have the following important proper-

i) Any solvable Lie algebra of dimension r contains an invariant subalgebra of dimension 1, which is contained in an invariant subalgebra of dimension 2 etc., i.e., r independent elements U1 , U2 , . . . , Ur may be found such that [Ui , Uk ] =

j=i X

cji,k Uj f or i = 1, . . . , r,

k = i + 1, . . . , r.

j=1

ii) The characteristic polynomial of a solvable algebra decomposes into linear factors, i.e., ∆(ω) = −ω

ρ=r i=r−1 Y  X i=1

 Eρ ciρi − ω .

ρ=i−1

Semi-simple and simple Lie algebras. If an algebra L is not solvable it may have invariant subalgebras that are solvable. There is always a unique largest solvable ideal which contains every solvable ideal, it is called the radical R. If R = L the algebra is solvable, if R = 0 the algebra does not have any nontrivial solvable ideal and the following definition applies. Definition 3.7 (Semi-simple and simple) If a nonsolvable Lie algebra does not have any nontrivial solvable invariant subalgebra it is called semi-simple. If it does not have any invariant subalgebra at all it is called simple.

120 From the above discussion it follows that determining the radical of any Lie algebra is an important step for identifying its structure. The following result which is due to Cartan [25], Chapter III, Section 5, gives a constructive answer to this problem. Theorem 3.10 (Cartan 1894) If L is a Lie algebra of dimension r with structure constants ckij and characteristic polynomial (3.17), its radical is determined by a linear combination of the basis elements the coefficients of which Pr ∂ψ (E) = 0. obey the constraints ρ=1 cρij 2 ∂Eρ Because ψ2 is a quadratic form, the constraints given in this theorem generate a system of linear homogeneous equations in the Ei , its solution determines the radical. The next theorem provides a constructive means for deciding semi-simplicity. Theorem 3.11 A Lie algebra is semi-simple if and only if the discriminant of the coefficient ψ2 (E) of the characteristic polynomial is different from zero. One of the major achievements of Killing and Cartan was to provide an explicit and complete classification of simple complex Lie algebras. It comprises four series of so-called classical algebras An , Bn , Cn and Dn for n ≥ 1, 2, 3 or 4 respectively, and the exceptional algebras E6 , E7 , E8 , F4 and G2 . The subindex denotes the rank of the respective algebra. Details may be found in the literature quoted at the beginning of this section. The importance of this classification becomes obvious from the following theorem which is also due to Killing and Cartan. Theorem 3.12 Any semi-simple Lie algebra is the direct sum of invariant simple subalgebras. The decomposition ensured by this theorem is unique. The simple components may be transformed into canonical form. In order to obtain this representation explicitly for any given semi-simple Lie algebra suitable algorithms are required. A lot of information on these algorithmic problems may be found in the thesis of de Graaf [37] and the references given there; see also the article by Cartan [26] on this subject. Example 3.5 the vector fields

(Cartan 1896) Let a six-dimensional Lie algebra given by

U1 = ∂x + x(x∂x + y∂y + z∂z ), U2 = ∂y + x(x∂x + y∂y + z∂z ), U3 = ∂z + x(x∂x + y∂y + z∂z ),

(3.18)

U4 = z∂y − y∂x , U5 = x∂z − z∂z , U6 = y∂x − x∂y . The coefficient ψ2 (E) = E12 + E22 + E32 + E42 + E52 + E62 has nonvanishing discriminant, consequently by Theorem 3.11 the Lie algebra is semi-simple.

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121

¯1 = U1 + U4 , U ¯2 = U2 + U5 , U ¯3 = U3 + U6 and It is easy to see that U ¯ ¯ ¯ U4 = U1 − U4 , U5 = U2 − U5 , U6 = U3 − U6 generate two type l3,1 algebras ¯i , U ¯i+3 ] = 0 for i = 1, 2, 3, i.e., the vector fields (3.18) generate a with [U semi-simple algebra which is the direct sum of two three-dimensional simple algebras of type l3,1 . The Levi Decomposition. A Lie algebra L having a nontrivial proper radical may be represented as a semidirect sum R >
S where R is the radical and S is semi-simple. The radical is unique and basis independent, the complement S is ismorphic to L/R. The formulation given in the subsequent theorem is taken from Rand [151]. Theorem 3.13 (Rand 1988) Let L be a Lie algebra with basis l1 , . . . , lr , radical R and Levi decomposition R >
S. There is an algorithm that finds a new basis r1 , . . . , rρ , s1 , . . . , sσ with ρ + σ = r such that R is generated by the ri , S by the sk and the commutation relations [R, R] ⊂ R, [S, S] = S and [R, S] ⊆ R are valid. The proof of Rand [151] provides a constructive method for obtaining such a basis explicitly. It is not given here because the examples that are relevant later on in this book are simple enough such that it may be obtained by inspection, and the general method is not required. Example 3.6 Consider the Lie algebra g23 (r = 3) defined on page 139. It is generated by the vector fields U1 = ∂x , U2 = ∂y , U3 = x∂x , x∂y , U4 = x∂y , U5 = y∂y , U6 = x2 ∂y , U7 = x2 ∂x + 2xy∂y from which the nonvanishing structure constants c113 = 1, c214 = 1, c317 = 2, c517 = 2, c225 = 1, c427 = 2, c636 = 2, c737 = 1, c445 = 1, c647 = 1, c656 = −1 and the coefficient ψ2 = 8E1 E7 + 3E52 − 6E3 E5 + E32 are obtained. Applying Theorem 3.10 yields the constraints E1 = E3 = E7 = 0, i.e., the radical is generated by U2 , U4 , U5 and U6 . It is easy to see that U1 , U3 + U5 and U7 generate a simple three-dimensional algebra. Consequently, the Levi decomposition has the form l4,6 (a = b = 1) >
l3,1 or, its equivalent in terms of vector fields {∂y , x∂y , x2 ∂y , y∂y } >
{∂x , x∂x + y∂y , x2 ∂x + 2xy∂y }.

3.3

Group Actions in the Plane

For the main subject of this book Lie transformation groups acting on a two-dimensional manifold are most important. If not stated otherwise this will

122 be the complex plane C. Therefore the results described in the two preceding sections are specialized to this case and illustrated by numerous examples, usually taken from the collection in Section 3.4. Of particular importance for later applications are constructive methods for dealing with these groups and its Lie algebras. Finite Groups and Their Infinitesimal Generators. The action of an r parameter Lie transformation group in coordinates x and y is written as x ¯ = f (x, y, a1 , . . . , ar ),

y¯ = g(x, y, a1 , . . . , ar )

(3.19)

with f (x, y, 0, . . . , 0) = x and g(x, y, 0, . . . , 0) = y. If there is only a single parameter, it is denoted by a without a subindex. Conforming with (3.7), the coefficients of the infinitesimal generators Uk = ξk (x, y)∂x + ηk (x, y)∂y are defined by ∂g ∂f and ηk (x, y) ≡ ξk (x, y) ≡ ∂ak a1 =...=ar =0 ∂ak a1 =...=ar =0 for k = 1, . . . , r. According to Lie’s First Theorem 3.1 there holds r

X ∂x ¯ = ϕj,k (a1 , . . . , ar )ξj (¯ x, y¯), ∂ak j=1

r

X ∂ y¯ = ψj,k (a1 , . . . , ar )ηj (¯ x, y¯). ∂ak j=1 (3.20)

Example 3.7

The rotations of the plane are parametrized by x ¯ = x cos α − y sin α,

y¯ = x sin α + y cos α

(3.21)

with 0 ≤ α < 2π. An additional second transformation of this type with parameter β ¯=x x ¯ cos β − y¯ sin β,

y¯ = x ¯ sin β + y¯ cos β

(3.22)

may be combined into a single one corresponding to a parameter value γ = α + β, the reverse operation of (3.21) corresponds to −α, and no action to α = 0. The infinitesimal generator is U = −y∂x + x∂y . The group equations d¯ y x = −¯ y and =x ¯. A Janet basis for the defining equations in (3.20) are d¯ dα dα  1 grlex, η > ξ, y > x term order is η + x y ξ = 0, ξx = 0, ξy + y ξ = 0 . Example 3.8 Assume that the points with coordinates (x, y) are transformed into (¯ x, y¯) according to x ¯ = x + a1 and y¯ = y + a2 . A second trans¯=x formation of the same type with parameters b1 and b2 yields x ¯ + b1 and y¯ = y¯ + b2 . Obviously the same effect may be achieved by a single transformation with parameter values c1 = a1 + b1 , c2 = a2 + b2 . The group equations ¯ = 1, ∂ x ¯ = 0, ∂ y¯ = 0 and ∂ y¯ = 1. For the infinitesimal (3.20) are ∂ x ∂a1 ∂a2 ∂a1 ∂a2 generators there follows U1 = ∂x and U2 = ∂y with [U1 , U2 ] = 0, i.e., they generate the Lie algebra l2,2 defined on page 135.

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123

In general a pair of substitution functions f and g does not define a group as the following example shows. Example 3.9

The transformations

x ¯ = (a + 1)x,

y¯ = y + a

and

¯ = (b + 1)¯ x x,

y¯ = y¯ + b

(3.23)

have the combined action ¯ = (a + 1)(b + 1)x, x

y¯ = y + a + b.

In this case it is obvious from the finite group equations that there does not exist a parameter c that may be expressed in the form c = φ(a, b). Example 3.10 An example of a group depending on three parameters is generated by the motions in the plane. If two subsequent motions are defined by x ¯ = x cos α − y sin α + a, y¯ = x sin α + y cos α + b, ¯=x x ¯ cos β − y¯ sin β + c,

y¯ = x ¯ sin β + y¯ cos β + d

a simple calculation shows that the combined motion is of the same form and is generated by the parameter values α + β,

a cos β + b sin β + c,

a sin β + b cos β + d

for the rotation angle and the two translations. The infinitesimal generators corresponding to the parameters a, b and α respectively are U1 = ∂x , U2 = ∂y and U3 = y∂x − x∂y with the nonvanishing commutators [U1 , U3 ] = −U2 and [U2 , U3 ] = U1 . Example 3.11 Consider the equations x ¯ = xea3 + a1 , y¯ = yea3 + a2 . By ∂ y¯ ∂ y¯ ∂ x ¯ ∂ x ¯ differentiation there follows = 1, = xea3 , = 1 and = yea3 ∂a1 ∂a3 ∂a2 ∂a3 for the nonvanishing derivatives. Consequently, corresponding to (3.20)   10 0 φ = ψ = 0 1 0  0 0 ea3 and ξ1 = 1, ξ2 = 0, ξ3 = x, η1 = 0, η2 = 1 and η3 = y. They satisfy (3.20), i.e. the above expressions for x ¯ and y¯ define a three parameter group. Substituting a2 = a3 = 0 yields x ¯ = x + a1 , y¯ = y and U1 = ∂x , substituting a1 = a3 = 0 yields x ¯ = x, y¯ = y + a2 and U2 = ∂y , finally substituting a1 = a2 = 0 yields x ¯ = xea3 , y¯ = yea3 and U3 = x∂x + y∂y with the commutators [U1 , U2 ] = 0, [U1 , U3 ] = U1 and [U2 , U3 ] = U2 , i.e., they generate a Lie algebra of type l3,2 (c = 1). The finite transformations corresponding to an infinitesimal generator U = ξ(x, y)∂x + η(x, y)∂y may be obtained from Lie’s equations (3.14). In

124 coordinates x and y they have the form d¯ x = ξ(¯ x, y¯), da

d¯ y = η(¯ x, y¯). da

(3.24)

This is a system of ode’s for the two functions x ¯(x, y, a) and y¯(x, y, a) w.r.t. to the group parameter a, complemented by the initial conditions x ¯(a = 0) = x, y¯(a = 0) = y. The coordinates of the manifold M appear only as parameters. Although a solution for such a system always exists, in general it cannot be obtained in closed form. This is possible only if ξ(¯ x, y¯) and η(¯ x, y¯) have a special structure. The following examples illustrate these possibilites. d¯ y x =x ¯2 , =x ¯y¯. Integration Let U = x2 ∂x + xy∂y , i.e., d¯ da da 1 of the first equation yields x ¯=− where φ is an undetermined funca + φ(x, y) 1 tion of its arguments. The requirement x ¯(a = 0) = x leads to φ(x, y) = − x x and x ¯ = 1 − ax . Substituting this value into the second equation yields ψ(x, y) d¯ y x¯ y = 1 − ax from which there follows y¯ = ax − 1 with ψ again an undeterda mined function of x and y. The condition y¯(a = 0) = y leads to ψ(x, y) = −y y and finally y¯ = 1 − ax . Example 3.12

d¯ y x =x Example 3.13 Let U = (x + y)∂x , i.e., d¯ ¯ + y¯, = 0. The last da da equation leads immediately to y¯ = y. Substitution into the first equation x −x yields the linear first order equation d¯ ¯ = y with the general solution x ¯= da a a e (ye + φ(x, y)). The initial condition x ¯(a = 0) = x leads to φ(x, y) = x − y, and the final answer x ¯ = ea x + ea (ea − 1)y. y x = y¯2 + 1, d¯ Example 3.14 Let U = (y 2 + 1)∂x + (x2 + 1)∂y , i.e., d¯ = da da x ¯2 + 1. Eliminating x ¯, the second order equation y¯002 − 4(¯ y 0 − 1)(¯ y 2 + 1) = 0 is obtained for which a closed-form solution cannot be found. For a selection of frequently occurring generators the corresponding finite transformation is given in Table 3.1. In order to find the finite transformations of an r-parameter group if the transformations corresponding to the infinitesimal generators are known, a method described in Lie [114], Kapitel 7, §3 will be applied. Lemma 3.2 (Lie 1893) Let r one-parameter groups be determined by the transformation functions fi (x, y, ai ) and gi (x, y, ai ) for i = 1, . . . , r. Then the general transformation of an r-parameter group is obtained recursively by x ¯ = fr (fr−1 (. . . f2 (f1 (x, y, a1 ), a2 ), . . . , ar−1 ), ar ), y¯ = gr (gr−1 (. . . g2 (g1 (x, y, a1 ), a2 ), . . . , ar−1 ), ar ).

(3.25)

Example 3.15 Let a three-parameter group be determined by the infinitesimal generators U1 = ∂x , U2 = ∂y , U3 = x∂x + y∂y . By means of the

Lie Transformation Groups

125

Generator ∂x

T ransf ormation x ¯ = x + a, y¯ = y

Generator ∂y

T ransf ormation x ¯ = x, y¯ = y + a

x∂x

x ¯ = xea , y¯ = y

y∂x

x ¯ = x + ay, y¯ = y

x∂y

x ¯ = x, y¯ = y + ax

y∂y

x ¯ = x, y¯ = yea

y∂x − x∂y

x ¯ = x cos a + y sin a, y∂x + x∂y x ¯ = x cosh a + y sinh a, y¯ = −x sin a + y cos a y¯ = x sinh a + y cosh a y y x 2 2 x ∂x + xy∂y x ¯ = 1 − ax , y¯ = 1 − ax xy∂x + y ∂y x ¯ = 1 −x ay , y¯ = 1 − ay y x2 ∂x + 2xy∂y x ¯ = 1 −xax , y¯ = (x + y)∂x x ¯ = ea x + (ea − 1)y, y¯ = y (1 − ax)2 y x∂x + (x + y)∂y x ¯ = ea x, y¯ = ea (ax + y) x2 ∂x + y 2 ∂y x ¯ = 1 −xax , y¯ = 1 − ay ex ∂y x ¯ = x, y¯ = y + aex mx∂x + ny∂y x ¯ = xema , y¯ = yena y n ∂x

x ¯ = x + ay n , y¯ = y

xn ∂y

x ¯ = x, y¯ = y + axn

TABLE 3.1: The finite transformations for a selection of frequently occurring infinitesimal generators are listed; a is the group parameter, the value a = 0 corresponds to the identity. finite transformations given in Table 3.1 one gets x ¯ = x + a1 , y¯ = y for U1 , ¯¯ = ea3 x ¯, y¯¯ = ea3 y¯ from U3 . Combining the x ¯=x ¯, y¯ = y¯ + a2 from U2 and x three transformations and making an obvious change of variables, the finite transformations of the three-parameter group are obtained in the form x ¯ = (x + a1 )ea3 , y¯ = (y + a2 )ea3 . The final result must be independent of the order in which the infinitesimal generators are applied. Taking for example U1 = x∂x + y∂y , U2 = ∂x and U3 = ∂y one obtains by a similar calculation x ¯ = eb1 x + b2 , y¯ = eb1 y + b3 . This is the same group with a different set of parameters. They are related by the equations b1 = a3 , b2 = a1 ea3 , b3 = a2 ea3 . On page 112 a third representation for a Lie transformation group in terms of the so-called defining equations has been introduced. Solving these linear homogeneous systems of pde’s is the main issue of Chapter 2. The reverse process, i.e., setting up the defining equations from a given set of infinitesimal generators, has also been discussed there on page 75. The algorithm ConstructJanetBasis has been designed for this purpose. The following simple example may be performed by pencil and paper, yet it shows all basic steps. Example 3.16 Let a group be given in terms of three vector fields by {∂x , ∂y , y∂x −x∂y }, consequently ξ = c1 +c3 y and η = c2 −c3 x. Differentiation

126 leads to ξx = 0, ξy = c3 , ηx = −c3 and ηy = 0. Eliminating c3 there follows ξx = 0, ξy + ηx = 0, ηy = 0. It is not a Janet basis. In a grlex term ordering with η > ξ, y > x the integrability condition ξyy = 0 is obtained. It has to be added to this system, i.e., the desired Janet basis is {ξx , ηx + ξy , ηy , ξyy }. The importance of the defining equations arises from the fact that the Lie symmetries of an ode are always obtained in this form in the first place, therefore these equations are called determining system in this context. In Section 3.5 it will be shown how many important properties of Lie groups may be obtained from these defining equations without solving them. Geometric Properties. Let x ¯ = f (x, y, a) and y¯ = g(x, y, a) be the finite equations of a one-parameter group with coordinates (x, y) and infinitesimal generator U = ξ(x, y)∂x + η(x, y)∂y . Under the action of this group an arbitrarily chosen point describes a curve the points of which are parametrized by a. Any two points on an individual curve may be transformed into each other by a suitable group element, they are equivalent w.r.t. this group. If they are located on different curves such a transformation does not exist. These curves, generated by the action of a one-parameter group on an arbitrary point of the plane, are called its path curve. They are special cases of the orbits defined on page 108. The path curves generate an invariant decomposition of the plane. By definition, the path curves corresponding to U obey equations (3.24). According to Theorem 2.19 these ode’s with first integral ϕ(x, y) = const are equivalent to the pde U ϕ = 0. It shows that the orbits of a one-parameter group of the plane are essentially determined by the invariant of the pde corresponding to its generator. Example 3.17 Consider the one-parameter group x ¯ = 1 −xax and y¯ = y y 2 1 − ax with generator U = x ∂x + xy∂y . The invariant of U is φ = x . y Consequently, its orbits are x = C, i.e., the straight lines through the origin. Example 3.18 The group considered in the preceding example had the y y¯ invariant φ = x . Substituting the transformed variables x ¯ and y¯ yields x ¯ = y 1 − ax y = x , i.e., the invariance of φ is explicit. 1 − ax x Example 3.19 The invariant of the one-parameter group with generator a 1 dx = 1 dy with the result φ = y . U = ax∂x + by∂y is determined by a x y b xb If the finite transformations of the group are known, deciding transitivity comes down to eliminating the group parameters a1 , . . . , ar from equations (3.19) for given (x, y) and (¯ x, y¯). Consider for example the group x ¯ = x + a1 , y¯ = y + a2 . If (¯ x, y¯) is considered to be chosen at will, the parameter values a1 = x ¯ − x, a2 = y¯ − y will perform the desired transformation. On the other hand, if the group is x ¯ = x, y¯ = y + a1 x + a2 , such a transformation does not

Lie Transformation Groups

127

exist because the transformations of the group leave x unchanged and a point with x ¯ 6= x can never be reached. The above question may be generalized to any number of points sharing this property. Example 3.20 The two-parameter groups g4 and g15 defined on page 138 are intransitive because the first coordinate is transformed as x ¯ = x, i.e., if x ¯ is chosen different from x there is no group element that brings about the transformation. Group g25 has finite transformations x ¯ = ea2 , y¯ = (y + a1 )ea2 x¯ y−x ¯y ¯ and a2 = log x from which the expressions a1 = x for the group x ¯ parameters are obtained, i.e., the group is transitive. The same is true for the group g26 that has been considered before. Example 3.21 Consider the group g6 on page 138 with finite transformations x ¯ = (x + a1 )ea3 , y¯ = (y + a2 )ea4 . Let two different points P1 = (x1 , y1 ) and P2 (x2 , y2 ) be arbitrarily given. Applying a transformation of the group to them yields x¯i = (xi + a1 )ea3 , y¯i = (yi + a2 )ea4 for i = 1, 2. The four group parameters may be eliminated with the result a1 =

x2 x ¯1 − x1 x ¯2 y2 y¯1 − y1 y¯2 x ¯1 − x ¯2 y¯1 − y¯2 , a2 = , ea3 = , ea4 = . x ¯2 − x ¯1 y¯2 − y¯1 x1 − x2 y1 − y 2

x1 , y¯1 ), P¯2 = (¯ x2 , y¯2 ) a This means, for any given pairs P1 , P2 and P¯1 = (¯ transformation of g6 may be determined moving Pi to P¯i for i = 1, 2. The same procedure for three arbitrarily chosen points leads to an inconsistent set of equations. Consequently, g6 is 2-fold transitive. If the finite transformations of the group are not known, or if their expressions are rather complicated, the criterion for transitivity in terms of the infinitesimal generators of the group given on page 114 may be applied. It allows deciding transitivity for all groups that are listed on page 138 from its vector field coefficients with the following answer. Transitive groups: All primitive groups g1 , g2 and g3 , the groups g5 , g6 , g7 and g9 to g12 allowing two systems of imprimitivity, the groups g13 , g14 and g17 to g23 with a single system of imprimitivity and finally the groups g24 , g25 and g26 allowing a one-parameter family of systems of imprimitivity. Intransitive groups: g4 , g8 , g15 , g16 and g27 . The possible types of transitivity of the transformation groups of the plane are completely described by the following result due to Lie [112], part I, page 632. Theorem 3.14 A finite continuous group of a two-dimensional manifold is at most four-fold transitive. The only group with maximal transitivity is the eight-parameter projective group g3 . This discussion shows that a group on a two-dimensional manifold may allow a single invariant, and this occurs if and only if all generators are multiples of a single one. In particular this is true for any single-parameter group.

128 The transitive two-parameter group x ¯ = x + a1 , y¯ = y + a2 considered previously transforms any line y = αx + const. into y¯ = α¯ x + const0 . with the same value of α, i.e., the straight lines with fixed slope α are permuted among themselves by the group. According to the definition given on page 108 they form a system of imprimitivity. The number of systems of imprimitivity is an important distinguishing feature of any transformation group. For a Lie transformation group of the plane the complete answer has been obtained by Lie [114], page 342, as follows. Theorem 3.15 A Lie transformation group of the plane is either primitive, or it has exactly one or two systems of imprimitivity, or there is a system of imprimitivity depending on a parameter or a function of one variable. In general it is difficult to identify a system of imprimitivity by means of the finite transformations of a group. A criterion for primitivity in terms of the infinitesimal generators of the group turns out to be more convenient. It is obtained by the following considerations. A system of imprimitivity in the plane may be regarded as a set of curves depending on a parameter which in turn may be described as the solution set of a first order ode. If an equation with this property does exist, the group under consideration must be a symmetry of it. Analytically this is assured (see Chapter 5) if for the generic rank of the prolonged coefficient matrix there holds   ξ1 , η1 , ζ1        ξ2 , η2 , ζ2  rank =3 .. ..   . .       ξr , ηr , ζr where ζk = ηk,x + y 0 (ηk,y − ξk,x ) − y 02 ξk,y , Lie [114], page 208. The ζ 0 s are discussed in detail in Chapter 5. In later applications the following provides a particularly well-suited criterion in terms of the Janet basis of a group (Schwarz [164], see also Neumer [135]). Theorem 3.16 Let the infinitesimal generators of a symmetry group of the x − y−plane be of the form U = ξ∂x + η∂y where the coefficients solve the corresponding determining system. In order for a system of imprimitivity different from x = const. or y = const. to exist, an equation ηy ω + ηx − ηωy − ξy ω 2 − ξx ω − ξωx = 0

(3.26)

with a suitable ω ≡ ω(x, y) must be valid as a consequence of the determining system of this symmetry group. If x=const. or y=const. is a system of imprimitivity, (3.26) reduces to ξy = 0 or ηx = 0 respectively. Proof Let the desired systems of imprimitivity be described by the first order ode y 0 = ω(x, y). By definition, the solutions of this first order equation are permuted by the original symmetry group. Therefore it must allow it as

Lie Transformation Groups

129

a symmetry group. Analytically this is expressed in terms of the requirement U (1) (y 0 − ω) = 0 on the solution set of y 0 = ω which is equivalent to (3.26), see Theorem 5.2; ω = 0 corresponds to y 0 = 0 or y = const., by exchange of x and y the condition for x = const. is obtained. For Lie the number and type of systems of imprimitivity was an important tool in order to obtain a classification of the transformation groups allowed by various manifolds. The classification of the groups of C2 given later in this chapter is organized by its systems of imprimitivity. The following examples are taken from this listing on page 138. Example 3.22 The group g1 has the Janet basis {ηy +ξx , ξxx , ξxy , ξyy , ηxx } for its defining equations. Reducing the condition (3.26) w.r.t. to it yields the equations ω = 0, ω 2 = 0, ωx = 0, ωy = 0 and 1 = 0 which is obviously inconsistent. Therefore g1 is primitive. Example 3.23 The group g7 has the Janet basis {ξy , ηx , ηy − cξx , ξxx } for its defining equations. From the first two equations x = const. and y = const. are obvious systems of imprimitivity according to the above lemma. The same group occurs in Example 5.16 as symmetry group. Reducing (3.26) w.r.t. to the second Janet basis given there leads to the equations ωx = 0, ωy = 0 and (ω − 6)(ω + 1) = 0 from which the two systems of imprimitivity y − 6x = const and y + x = const are obtained.  Example 3.24 The group g13 has the Janet basis ξx − y2 η, ξy , ηy − y1 η, ηxx for its defining equations. The single system of imprimitivity x = const is obvious from the second equation, reduction of (3.26) w.r.t. to this Janet basis leads to an inconsistent system. In Example 5.14 the same group type generates the Janet basis in some transformed variables. Reduction of (3.26) w.r.t. the Janet basis given there leads to the system ωx +

1 2 1 1 2 1 ω2 − ω+ = 0, ωy + ω2 − ω+ =0 x+y x+y x+y x+y x+y x+y

and (ω − 1)2 = 0 with the unique solution ω = 1, i.e., the single system of imprimitivity is y − x = const. Similarity. Transformation to Canonical Form. The appearance of any Lie transformation group may be changed in two ways: Either by a change of the group parameters, leading essentially to a new basis of the Lie algebra generated by the infinitesimal generators, or by changing the coordinates of the transformed manifold, leaving the commutation relations between the infinitesimal generators unchanged according to Lemma 3.1, but changing the form of the generators as has been shown on page 111 for a n-dimensional manifold. Due to its importance for the rest of this chapter, it is given here explicitly for a two-dimensional manifold. Lemma 3.3

Let U = ξ(x, y)∂x + η(x, y)∂y be a vector field in the plane

130 with coordinates x and y. Furthermore, let new coordinates u and v be defined by u = σ(x, y) and v = ρ(x, y) with the inverse x = φ(u, v) and y = ψ(u, v). Then the coefficients of the transformed vector field V = α(u, v)∂u + β(u, v)∂v are β(u, v) = U ρ x=φ,y=ψ . α(u, v) = U σ x=φ,y=ψ , A constructive method for deciding similarity of two transformation groups defined in terms of their infinitesimal generators has been given in Theorem 2. It is illustrated next by a simple example. Example 3.25 Consider the generators U1 = ∂u , U2 = ∂v , U3 = u∂u + v∂v in coordinates u and v with commutators [U1 , U2 ] = 0, [U1 , U3 ] = U1 and [U2 , U3 ] = U2 . They form a Lie algebra of type l3,2 (c = 1). On the other 1 ∂ , V = − x2 ∂ + (x + y)∂ and hand, there are the generators V1 = y1 ∂x + x y 2 y y x
1 1 V3 = y ∂x + y + x ∂y in coordinates x and y with commutators [V1 , V2 ] = V1 , [V1 , V3 ] = V1 and [V2 , V3 ] = V2 − V3 . The linear combinations V¯1 = 12 V1 , V¯2 = 21 (V1 + V2 − V3 ) and V¯3 = V3 − V1 obey the same commutation as the generators Ui . Explicitly they are 1 1 V¯1 = ∂x + ∂y , 2y 2x

x2 x V¯2 = − ∂x + ∂y , 2y 2

V¯3 = y∂y .

y The relations U3 = uU1 + vU2 and V3 = xyV1 + x V2 lead to the coordinate y transformation u = xy and v = x . A special kind of similarity problem consists of the following question: Given the infinitesimal generators of a transformation group in some set of coordinates, is this group similar to another one which is also given in terms of its infinitesimal generators? This latter group is considered to be a canonical form chosen from a classification of groups like those described in Section 3.5. If similarity is assured, the transformation functions between the two coordinate sets may be determined. For a one-parameter group there are no commutation relations. The canonical form of the group is the translation in one of the coordinates as explained on page 112. A constructive solution to this problem is given next. Lemma 3.4 Let ξ(x, y)∂x + η(x, y)∂y be a vector field with coordinates x and y, and ∂v its canonical form in coordinates u = σ(x, y), v = ρ(x, y). i) If ξ 6= 0 and η 6= 0, the transformation functions are Z dx σ(x, y) = Φ(φ(x, y)), ρ(x, y) = Ψ(φ(x, y)) + ¯ |y¯=φ ξ(x, y = φ) dy η where φ(x, y) = const is the first integral of = , y¯ = φ(x, y) with dx ξ ¯ y¯). the inverse y = φ(x,

Lie Transformation Groups

131 Z

ii) If ξ = 0, η 6= 0 there holds σ = Φ(x), ρ = Ψ(x) +

dy . η(x, y)

Z

dx . In all cases ξ(x, y) Φ and Ψ are undetermined functions of its argument.

iii) If ξ 6= 0, η = 0 there holds σ = Φ(y), ρ = Ψ(y) +

Proof According to Lemma 3.3, because α = 1 and β = 0 now, σ and ρ satisfy the system ξσx + ησy = 0 and ξρx + ηρy = 1. The solution of this system is obtained from Corollary 2.7 on page 78. This lemma shows that in general it is not guaranteed that the canonical form ∂v for an arbitrary vector field may be obtained in closed form. The dy η crucial step is to solve the first order differential equation = . This is dx ξ only possible if the coefficients ξ(x, y) and η(x, y) are sufficiently simple. This remark will be important in Chapter 5 when symmetries of first order ode’s are searched for. Let 2x∂x + 3y∂y be the given vector field. The equation y2 y2
for σ is 2xσx + 3yσy = 0. Its invariant 3 yields σ = Φ 3 with Φ an x x undetermined function. The equation for ρ is 2xρx + 3yρy = 1, it has been considered in Example 2.57. If Φ = 0 is chosen in the solution for z given y2 there, ρ = − 21 log x is obtained. Consequently, canonical variables are u = 3 , x √ v = − 21 log x or x = e−2v , y = ue−3v . Example 3.26

Given an infinitesimal generator, the corresponding finite transformations may be obtained also by transforming it to canonical form first, and then substituting the backtransformation into the corresponding translation. In the next example this is shown for the group considered previously in Example 3.12. Example 3.27 Let the infinitesimal generator be U = x2 ∂x + xy∂y . The transformation functions σ(x, y) and ρ(x, y) satisfy x2 σx + xyσy = 0 and dy x2 ρx + xyρy = 1. By Corollary 2.7, the former equation leads to y = dx x y with the special solution φ ≡ x = C. Introducing x ¯ = φ as new variable, the integral for ρ yields Z ρ=

dy =x ¯ xy

Z

dy x ¯ 1 2 = −y = −x. y

y 1 The integration constant has been omitted. Substituting u = x and v = − x into the canonical transformation finally leads to y¯ y y 1 = − 1 + a or x ¯ = 1 +xax , y¯ = 1 + ax . x x ¯ = x, −x ¯

132 Generator x∂x

σ(x, y) y

y∂x

y y x xy

x∂x + y∂y x∂x − y∂y

ρ(x, y) log x x y log x − log y

φ(u, v) ψ(u, v) ev u uv

u

ev

uev

uev

e−v

a

y 1 log y u−1/b eav ebv b b p x y −y∂x + x∂y x2 + y 2 arctan x u cos v u sin v

ax∂x + by∂y

TABLE 3.2: For a selection of vector fields ξ(x, y)∂x +η(x, y)∂y the transformation functions x = φ(u, v), y = ψ(u, v) to canonical form ∂v , and the inverse functions u = σ(x, y), v = ρ(x, y) are given. It is useful to have the transformation functions to canonical form for a selection of vector fields occurring frequently in applications easily available. They are listed in Table 3.2. For groups with more than a single parameter, certain normal forms have been introduced by Lie. Let the generators be given as Ui = ξi (x, y)∂x + ηi (x, y)∂y for i = 1, . . . , r.

(3.27)

The determinants ∆ij = ξi ηj − ξj ηi and especially ∆12 ≡ ∆ will be used frequently. The canonical coordinates are u and v. The solution to the transformation problem is based on the expressions for the coefficients of the transformed generators given in Lemma 3.3, and the solution procedure for quasilinear systems of pde’s described in Lemma 2.12 of the preceding chapter. The operations required are determined above all by the dimension r of the group. For r = 2 the complete answer is given next. Lemma 3.5 Let ξi ∂x + ηi ∂y , i = 1, 2 be the generators of a two-parameter group satisfying the proper commutator for one of the groups below. The two transformation functions u = σ(x, y) and v = ρ(x, y) to canonical variables are determined by the following systems of equations where ∆ = ξ1 η2 − ξ2 η1 . Group g4 , {∂v , v∂v } : Constraint ∆ = 0. dx dy ξ2 = , solution φ(x, y) = C, σ ≡ σ(φ(x, y)), ρ = . ξ1 η1 ξ1 Group g15 , r = 2, {∂v , u∂v } : Constraint ∆ = 0. Z dy ξ2 σ= , ρ= . ξ1 η1 (σ(x, y), y) Group g26 , {∂u , ∂v }: Constraint ∆ 6= 0. σx =

η2 ξ2 η1 ξ1 , σy = − , ρx = − , ρy = . ∆ ∆ ∆ ∆

Lie Transformation Groups

133

Group g25 , {∂u , u∂u + v∂v }: Constraint ∆ 6= 0.

σx +

Proof

η2 ξ1 ξ2 η1 ξ1 η1 σ = , σy − σ = − , ρx + ρ = 0, ρy − ρ = 0. ∆ ∆ ∆ ∆ ∆ ∆

For the group g4 , Lemma 3.3 leads to the system

ξ1 σx + η1 σy = 0, ξ1 ρx + η1 ρy = 1, ξ2 σx + η2 σy = 0, ξ2 ρx + η2 ρy = ρ. The existence of a nontrivial solution for σ requires the constraint ∆ = 0. If it is satisfied, σ is obtained from the first equation. The expression for ρ follows from the consistency of the two equations for ρ. For the second group g15 the same system is obtained except that ρ at the right hand side of the last equation is replaced by σ. Again ∆ = 0 is required. The consistency for the equations determining ρx and ρy now leads to the expressions for σ; ρ has to be determined from ξ1 ρx + η1 ρy = 1. The corresponding homogeneous equation has σ as its solution which is already known. Consequently, the integral for ρ follows by Lemma 3.4. For g26 the system for σ and ρ is ξ1 σx + η1 σy = 1, ξ1 ρx + η1 ρy = 0, ξ2 σx + η2 σy = 0, ξ1 ρx + η2 ρy = 1, and for g25 ξ1 σx + η1 σy = 1, ξ1 ρx + η1 ρy = 0, ξ2 σx + η2 σy = σ, ξ2 ρx + η2 ρy = ρ. In either case algebraic elimination of the first partial derivatives yields the equations given above. Example 3.28 Consider the vector fields U1 = ∂x + ∂y , U2 = y∂x + x∂y satisfying [U1 , U2 ] = U1 generating a group of type g25 . According to the above lemma, the transformation functions σ and ρ obey σx +

x 1 y 1 σ= , σy − σ=− , x−y x−y x−y x−y ρx +

1 1 ρ = 0, ρy − ρ = 0. x−y x−y

Its solution is obtained by Lemma 2.9 and Lemma 2.10 with the result σ = x+y C1 C2 x − y + 2 and ρ = x − y . Choosing C1 = C2 = 1 yields u ≡ σ(x, y) = x+y 1 1 x − y + 2 and v ≡ ρ(x, y) = x − y , and the inverse transformations 1 and y ≡ ψ(u, v) = u − v − 1 . x ≡ φ(u, v) = u − v + 2v 2v Differential Invariants. In the plane there is only a single dependency possible between the coordinates if no additional variables are introduced. Let the coordinates again be x and y and assume that y depends on x, i.e.,

134 y ≡ y(x). This case covers ordinary differential equations in a single dependent variable. The next two examples deal with invariants of this type. Example 3.29 Consider the vector fields U1 = ∂x , U2 = 2x∂x + y∂y and U3 = x2 ∂x + xy∂y obeying [U1 , U2 ] = 2U1 , [U1 , U3 ] = U2 and [U2 , U3 ] = 2U3 . If y ≡ y(x) is assumed, its second prolongations are (2)

U1

(2)

= U1 , U2

(2)

= U2 − y 0 ∂y0 − 3y 00 ∂y00 , U3

= U3 − (xy 0 − y)∂y0 − 3xy 00 ∂y00 .

A simple calculation shows that the first two commutators of the prolonged vector fields remain unchanged. The last one is obtained from (2)

(2)

U2 U3

= 4x2 ∂x + 3xy∂y − (xy 0 − y)∂y0 + 3xy 00 ∂y00

(2)

= 2x2 ∂x + xy∂y + (xy 0 − y)∂y0 + 9xy 00 ∂y00

and (2)

U3 U2 (2)

(2)

with the result U2 U3

(2)

(2)

− U3 U2

(2)

= 2U3 .

Example 3.30 Consider the four-parameter group generated by ∂x , ∂y , x∂y and x∂x + y∂y . If it is assumed that y ≡ y(x), the fifth prolongations are ∂x , ∂y , x∂y + ∂y0 and x∂x + y∂y − y 00 ∂y00 − 2y 000 ∂y000 − 3y (4) ∂y(4) − 4y (5) ∂y(5) . y 003 y 002 By the methods explained in Chapter 2 the invariants Φ1 = 000 , Φ2 = (4) y y y 004 and Φ3 = (5) are obtained. The invariants Φ1 and Φ2 are a full system. y More general types of differential invariants for transformation groups in the plane may be obtained if dependencies on variables different from the coordinates are assumed. In the next example the group is the same as in the preceding one, the coordinates are now v and w, they are assumed to depend on x and y. Example 3.31 Consider the four-parameter group generated by ∂v , ∂w , v∂w and v∂v + w∂w . It is assumed now that both v and w depend on x and y. These latter variables are not transformed by the group. The second prolongations w.r.t. x and y are ∂v , ∂w , v∂w + vx ∂wx + vy ∂wy + vxx ∂wxx + vxy ∂wxy + vyy ∂wyy , v∂v + w∂w + vx ∂vx + wx ∂wx + vy ∂vy + wy ∂wy +vxx ∂vxx + wxx ∂wxx + vxy ∂vxy + wxy ∂wxy + vyy ∂vyy + wyy ∂wyy . v v w −v w There are two first order invariants Φ1 = vxy and Φ2 = x y 2 y x , and vx three second order invariants are v w −v w v w −v w Φ6 = vxx wx −2 vx wxx , Φ7 = yy y 2 y yy , Φ8 = xy x 2 x xy . vx vy vx

Lie Transformation Groups

3.4

135

Classification of Lie Algebras and Lie Groups

Lie’s classification of groups of the complex plane and of the complex Lie algebras of low dimension is described in this section. It is not explained how Lie arrived at these classifications, only references to the literature are given. Lie Algebras of Low Dimensions. For dimension not higher than four the listing of complex Lie algebras is complete. In some cases there occur parameters. They have to be constrained suitably in order to avoid duplication. A few algebras of higher dimension are also given because they are needed for the symmetry analysis later on. In addition the following general rule is applied: Whenever an essential algebraic property like for example the dimension of the derived algebra changes for a special value of a parameter, the algebra corresponding to these parameter values is listed as a separate entry. More details may be found in Lie’s original work [112], vol. III, or in vol. 41, Chapter 7 of the Encyclopedia of Mathematical Sciences. Each entry of the subsequent listing is a representative of a full isomorphism class. In order to identify a given Lie algebra, membership to a particular isomorphism class has to be decided. Secondly, the transformation to canonical form may be determined. Various properties of Lie algebras may be read off easily from this listing. For example, all two-dimensional algebras, all three-dimensional algebras except l3,1 and all four-dimensional algebras except l4,1 are solvable; l3,1 is simple; l4,1 has the Levi decomposition l1 >
l3,1 where the radical is generated by U4 and the simple three-dimensional algebra by the remaining generators. In the lisiting below only nonvanishing commutators are given. One-dimensional algebras l1 : commutative, Characteristic equation: ω = 0. Two-dimensional algebras l2,1 : [U1 , U2 ] = U1 , dim l02,1 = 1, dim l002,1 = 0. Characteristic equation: ω 2 − E2 ω = ω(ω − E2 ) = 0. l2,2 : commutative. Characteristic equation: ω 2 = 0. Three-dimensional algebras l3,1 : [U1 , U2 ] = U1 , [U1 , U3 ] = 2U2 , [U2 , U3 ] = U3 , dim l03,1 = dim l003,1 = 3. Characteristic equation: ω 3 + (4E3 E1 − E22 )ω = ω(ω 2 + 4E3 E1 − E22 ) = 0. l3,2 (c): [U1 , U3 ] = U1 , [U2 , U3 ] = cU2 , c 6= 0, dim l03,2 = 2, dim l003,2 = 0. Characteristic equation: ω 3 −(c+1)E3 ω 2 +cE32 ω = ω(ω −E3 )(ω −cE3 ) = 0. l3,3 : [U1 , U3 ] = U1 , [U2 , U3 ] = U1 + U2 , dim l03,3 = 2, dim l003,3 = 0. Characteristic equation: ω 3 − 2E3 ω 2 + E32 ω = ω(ω − E3 )2 = 0. l3,4 : [U1 , U3 ] = U1 , dim l03,4 = 1, dim l003,4 = 0. Characteristic equation: ω 3 − E3 ω 2 = ω 2 (ω − E3 ) = 0. l3,5 : [U2 , U3 ] = U1 , dim l03,5 = 1, dim l003,5 = 0. Characteristic equation: ω 3 = 0. l3,6 : commutative. Characteristic equation: ω 3 = 0.

136 Four-dimensional algebras l4,1 : [U1 , U2 ] = U1 , [U1 , U3 ] = 2U2 , [U2 , U3 ] = U3 , dim l04,1 = dim l004,1 = 3. Characteristic equation: ω 4 + (4E3 E1 − E22 )ω 2 = (ω 2 + 4E3 E1 − E22 )ω 2 = 0. l4,2 : [U1 , U4 ] = U1 , [U2 , U4 ] = U1 + U2 , [U3 , U4 ] = U2 + U3 , dim l04,2 = 3, dim l004,2 = 0. Characteristic equation: ω 4 − 3E4 ω 3 + 3E42 ω 2 − E43 ω = ω(ω − E4 )3 = 0. l4,3 (c): [U1 , U4 ] = cU1 , [U2 , U4 ] = (c + 1)U2 , [U3 , U4 ] = U1 + cU3 , dim l04,3 = 3, dim l004,3 = 0. Characteristic equation: ω 4 − (3c + 1)E4 ω 3 + c(3c + 2)E42 ω 2 − c2 (c + 1)E43 ω = ω(ω − cE4 )2 (ω − (c + 1)E4 ) = 0. l4,4 : [U1 , U4 ] = 2U1 , [U2 , U3 ] = U1 , [U2 , U4 ] = U2 , [U3 , U4 ] = U2 + U3 , dim l04,4 = 3, dim l004,4 = 1, dim l000 4,4 = 0. Characteristic equation: ω 4 − 4E4 ω 3 + 5E42 ω 2 − 2E43 ω = −ω(ω − 2E4 )(ω − E4 )2 = 0. l4,5 (c): [U1 , U4 ] = cU1 , [U2 , U3 ] = U1 , [U2 , U4 ] = U2 , [U3 , U4 ] = (c − 1)U3 , c 6= 1, dim l04,5 = 3, dim l004,5 = 1, dim l004,4 = 0. Characteristic equation: ω 4 − 2cE4 ω 3 + (c2 + c − 1)E42 ω 2 − c(c − 1)E43 ω = ω(ω − E4 )(ω − cE4 )(ω − (c − 1)E4 )) = 0. l4,6 (a, b): [U1 , U4 ] = U1 , [U2 , U4 ] = aU2 , [U3 , U4 ] = bU3 , ab 6= 0, dim l04,6 = 3, dim l004,6 = 0. Characteristic equation: ω 4 − 2E2 ω 3 + E42 ω 2 = ω 2 (ω − E4 )2 = 0. l4,7 : [U1 , U4 ] = U1 , [U2 , U4 ] = U2 , [U3 , U4 ] = U2 + U3 , dim l04,7 = 2, dim l004,7 = 0. Characteristic equation: ω 4 − 3E4 ω 3 + 3E42 ω 2 − E43 ω = ω(ω − E4 )3 = 0. l4,8 : [U1 , U2 ] = U2 , [U1 , U3 ] = U2 + U3 , dim l04,8 = 2, dim l004,8 = 0. Characteristic equation: ω 4 + 2E1 ω 3 + E12 ω 2 = ω 2 (ω + E1 )2 = 0. l4,9 (a): [U1 , U2 ] = U2 , [U1 , U3 ] = aU3 , a 6= 0, dim l04,9 = 2, dim l004,9 = 0. Characteristic equation: ω 4 − (a + 1)E4 ω 3 + aE42 ω 2 = ω 2 (ω − E4 )(ω − aE4 ) = 0. l4,10 : [U1 , U2 ] = U2 , [U3 , U4 ] = U4 , dim l04,10 = 2, dim l004,10 = 0. Characteristic equation: ω 4 + (E3 + e1 )ω 3 + E3 E1 ω 2 = ω 2 (ω + E1 )(ω + E3 ) = 0. l4,11 : [U2 , U4 ] = U2 , [U3 , U4 ] = U1 , dim l04,11 = 2, dim l004,11 = 0. Characteristic equation: ω 4 − E4 ω 3 = ω 3 (ω − E4 ) = 0. l4,12 : [U1 , U4 ] = U1 , [U2 , U3 ] = U1 , [U2 , U4 ] = U2 , dim l04,12 = 2, dim l004,12 = 0. Characteristic equation: ω 4 − 2E4 ω 3 + E42 ω 2 = ω 2 (ω − E4 )2 = 0. l4,13 : [U1 , U4 ] = U2 , [U3 , U4 ] = U1 , dim l04,13 = 2, dim l004,13 = 0. Characteristic equation: ω 4 = 0. l4,14 : [U1 , U4 ] = U1 , [U2 , U4 ] = U2 , dim l04,14 = 2, dim l004,14 = 0. Characteristic equation: ω 4 − 2E2 ω 3 + E42 ω 2 = ω 2 (ω − E4 )2 = 0. l4,15 : [U1 , U4 ] = U1 , dim l04,15 = 1, dim l004,15 = 0. Characteristic equation: ω 4 − E4 ω 3 = ω 3 (ω − E4 ). l4,16 : [U1 , U2 ] = U3 , dim l04,16 = 1, dim l004,16 = 0. Characteristic equation: ω 4 = 0.

Lie Transformation Groups

137

l4,17 : commutative. Characteristic equation: ω 4 = 0. In order to faciliate the comparison with the classification as given by Lie [112], vol. III, §136 and §137, in the subsequent table the Lie algebras l4,i and the corresponding equation number of Lie are given without the parameter constraints. i in l4,i

1 2 3 4 5 6 7 10 12 13 16 17

0

Lie s enumeration 58 70 68 63 62 67 64 65 69 71 72 73 Algebra l4,8 is obtained from l4,3 for the parameter value c = −1, the permutations U2 and U4 , U1 and U2 and the replacement U2 by −U2 . Algebra l4,9 is obtained from l4,6 for the parameter value b = 0. Algebra l4,11 is obtained from l4,3 for the parameter value c = 0. Algebra l4,14 is obtained from l4,6 for the parameter values a = 1, b = 0. Algebra l4,15 is obtained from l4,6 for the parameter values a = b = 0. Some algebras with more than four dimensions l8 : [U1 , U3 ] = U1 , [U1 , U5 ] = U2 , [U1 , U7 ] = U4 , [U1 , U8 ] = U6 + 2U3 , [U2 , U4 ] = U1 , [U2 , U6 ] = U2 , [U2 , U7 ] = 2U6 + U3 , [U2 , U8 ] = U5 , [U3 , U4 ] = −U4 , [U3 , U5 ] = U5 , [U3 , U8 ] = U8 , [U4 , U5 ] = U6 − U3 , [U4 , U8 ] = U7 , [U5 , U6 ] = U5 , [U6 , U7 ] = U7 . Characteristic equation: The two leading coefficients of (3.17) are ψ1 = −2E6 ,

ψ2 = 3E2 E7 + 6E1 E8 + E62 + E6 E3 − 2E32 .

Finite Transformation Groups. These are the most important groups occurring in this book because any group of Lie symmetries of an ode must be similar to one of them. As it will turn out later on, however, not all of them do actually occur as a symmetry group of an ode of low order. These questions will be considered in Chapter 5. The subsequent classifications correspond to a manifold C2 with coordinates x and y as given by Lie [112], vol. III, Kapitel 3, pages 28-78. In the real plane R2 the classification of primitive groups is different, it is given in the same book on pages 360-392. A more complete discussion may be found in Gonzales-Lopez [54], and in Chapter 6 of vol. I of [43]. A partial ordering for these groups has been given by Krause [96], page 260. There are certain special classes of groups that have played an important role for the development of the subject, the so-called linear groups. They are subgroups of the group GL(V ) of linear transformations of a finite-dimensional vector space over the field R or C. On the one hand they were the first examples of such groups investigated by Lie. Secondly, their structure turned out to be fundamental in the sense that other groups may be composed out of them by standard operations like products or semidirect products. These classes are described first.

138 i) The n(n + 2)-parameter projective group ak,1 x1 + ak,2 x2 + . . . + ak,n xn + ak,n+1 xn+1 an+1,1 x1 + an+1,2 x2 + . . . + an+1,n xn+1 P with generators ∂xi , xi ∂xk , xi xj ∂xj , i, j, k = 1, . . . , n. x ¯k =

ii) The n(n + 1)-parameter general linear group x ¯k = ak,1 x1 + ak,2 x2 + . . . + ak,n xn + ak,n+1 with generators ∂xi , xi ∂xk . iii) The n2 -parameter general linear homogeneous group x ¯k = ak,1 x1 + ak,2 x2 + . . . + ak,n xn with generators xi ∂xk , i, k = 1, . . . , n. iv) The n(n+1)−1-parameter special linear group of those transformations in ii) with determinant equal to 1. Its generators are ∂xi , xi ∂xk and xi ∂xi − xk ∂xk , i, k = 1, . . . , n, i 6= k. v) The n2 − 1-parameter special linear homogeneous group of those transformations in iii) with determinant equal to 1. Its generators are xi ∂xk , xi ∂xi − xk ∂xk , i, k = 1, . . . , n, i 6= k. A systematic notation for the groups defined in ii), iii), iv) and v) is GLn , SLn , GLHn and SLHn . However, as van der Waerden [186], page 5, has pointed out, it became popular to use the notation GLn and gln , SLn and sln for the homogeneous groups iii) and v) that will be applied in this book. For n = 2, using the variables x and y, their explicit form is gl2 ≡ {x∂x , y∂x , x∂y , y∂y }, type l4,1 , sl2 ≡ {x∂y , y∂x , x∂x − y∂y }, type l3,1 . The problem of determining a basis for gl2 and sl2 such that the commutator table given in the above listing applies is considered in Exercise 3.11. Each entry of the subsequent listing is a representative of a group type, i.e., the totality of all groups that are similar to this entry by a diffeomorphism. In order to identify any given group, its type has to be found. After that the transformation to the representative in this list may be determined. Primitive groups g1 : {∂x , ∂y , x∂y , x∂x − y∂y , y∂x }. g2 : {∂x , ∂y , x∂y , y∂y , x∂x , y∂x }. g3 : {∂x , ∂y , x∂y , y∂y , x∂x , y∂x , x2 ∂x + xy∂y , xy∂x + y 2 ∂y }. Groups with two systems of imprimitivity x = const and y = const. g4 : {∂y , y∂y }.

Lie Transformation Groups

139

g5 : {∂x , ∂y , y∂y }. g6 : {∂x , ∂y , y∂y , x∂x }. g7 : {∂x , ∂y , x∂x + cy∂y } with c 6= 0, 1. g8 : {∂y , y∂y , y 2 ∂y }. g9 : {∂x , ∂y , y∂y , y 2 ∂y }. g10 : {∂x + ∂y , x∂x + y∂y , x2 ∂x + y 2 ∂y }. g11 : {∂x , ∂y , y∂y , x∂x , y 2 ∂y }. g12 : {∂x , ∂y , y∂y , x∂x , y 2 ∂y , x2 ∂x }. Groups with the system of imprimitivity x = const g13 : {∂x , 2x∂x + y∂y , x2 ∂x + xy∂y }. g14 : {y∂y , ∂x , x∂x , x2 ∂x + xy∂y }. g15 : {φ1 (x)∂y , φ2 (x)∂y , . . . , φr (x)∂y } with r ≥ 2. Size: r ≥ 2. g16 : {φ1 (x)∂y , φ2 (x)∂y , . . . , φr (x)∂y , y∂y } with r ≥ 2. Size: r + 1 ≥ 3. g17 :P {eαk x ∂y , xeαk x ∂y , . . . , xρk eαk x ∂y , ∂xP } with l ≥ 1, 1 ≤ k ≤ l, ρk + l ≥ 2, α1 (α1 − 1) = 0. Size: ρk + l + 1 ≥ 3. kx g18 : {eαP ∂y , xeαk x ∂y , . . . , xρk eαk x ∂y , y∂P y , ∂x } with l ≥ 1, 1 ≤ k ≤ l, l + ρk ≥ 2, α1 = 0, α2 = 1. Size: ρk + l + 2 ≥ 4. g19 : {∂y , x∂y , . . . , xr−1 ∂y , ∂x , x∂x + cy∂y } with r ≥ 2. Size: r + 2 ≥ 4. g20 : {∂y , x∂y , . . . , xr−1 ∂y , ∂x , x∂x +(ry +xr )∂y } with r ≥ 1. Size: r +2 ≥ 3. g21 : {∂y , x∂y , . . . , xr−1 ∂y , y∂y , ∂x , x∂x } with r ≥ 2. Size: r + 3 ≥ 5. g22 : {∂y , x∂y , . . . , xr−1 ∂y , ∂x , 2x∂x + (r − 1)y∂y , x2 ∂x + (r − 1)xy∂y } with r ≥ 2. Size: r + 3 ≥ 5. g23 : {∂y , x∂y , . . . , xr−1 ∂y , y∂y , ∂x , x∂x , x2 ∂x + (r − 1)xy∂y } with r ≥ 2. Size: r + 4 ≥ 6. Groups with the system of imprimitivity y = αx + const g24 : {∂x , ∂y , x∂x + y∂y }. g25 : {∂y , x∂x + y∂y }. g26 : {∂x , ∂y }. Groups with the system of imprimitivity y = φ(x) + const g27 : {∂y }. This listing of the finite groups of the complex plane follows most closely the one given in Lie [112], vol. III. For the indicated range of parameters the listing is complete without any duplication. In earlier publications Lie gave some of these groups in slightly different form, compare the discussion in Lie [112], vol. III, page 76. Group g7 with c 6= 1 is listed separately from group g24 because the value c = 1 in the latter group generates a different system of imprimitivity than for c 6= 1, and for the value c = 0 it is similar to g5 . A further distinction made by Lie concerns the multiplicity of a system of imprimitivity. It generates the additional distinctions as follows. For group g22 with r = 3 the system x = const. counts twofold (last line on page 71), for r = 2 and r ≥ 4 it counts as a single system (first item after the heading on page 72: b) Die invariante Schar z¨ ahlt doppelt). For group g20 and r = 1

140 the system is counted twofold (last line on page 72), all remaining cases r > 1 are single counted (item six from top of page 72). Finally the imprimitivity system for group g19 with c = 1 counts twofold (second line from bottom of page 72) and only single for the remaining cases c 6= 1 (line five from top of page 72). These remarks allow it to identify each group in Lie’s listing uniquely. Some applications require the knowledge of all group types with a given number of parameters without duplication. For most groups of the above listing it is easy to identify them. The coefficients of ∂y for the groups g17 and g18 are obtained by the following observations. The desired coefficients are solutions of a linear homogeneous ode with constant coefficients of the proper order (Lie [112], vol. III, page 54ff). As a consequence they have the general form xk eαx . Subsequently the infinitesimal generators of various groups of a given size r up to r ≤ 7 are listed. For r ≤ 4 this listing is complete. For values higher than four only those groups are listed that occur later on as symmetry groups of differential equations. In addition the Janet bases for the defining equations are given in grex term ordering with η > ξ and y > x. The coefficients φk for k = 1, . . . , r occurring in the group types g15 and g16 are undetermined functions of x. The coefficients qk occurring in the Janet basis for these groups are defined by (2) W (φ1 , φ2 ) qk = − k(2) W (φ1 , φ2 ) for k = 1, . . . , r, see eq. (2.4). The algebraic properties of groups with r ≤ 4 may be obtained from their Lie algebra type which is always given. One-parameter group g27 : {∂y }. Type l1 . Janet basis {ξ, ηx , ηy }. Two-parameter groups g4 : {∂y , y∂y }. Type l2,1 . Janet basis {ξ, ηx , ηyy }. g15 (r = 2): {φ1 ∂y , φ2 ∂y }. Type l2,2 . Janet basis {ξ, ηy , ηxx + q1 ηx + q2 η}.  g25 : {∂x , x∂x + y∂y }. Type l2,1 . Janet basis ξx − y1 η, ξy , ηx , ηy − y1 η . g26 : {∂x , ∂y }. Type l2,2 . Janet basis {ξx , ξy , ηx , ηy }. Three-parameter groups g5 : {∂x , ∂y , y∂y }. Type l3,4 . Janet basis {ξx , ξy , ηx , ηyy }. g7 : {∂x , ∂y , x∂x + cy∂y } with c 6= 0, 1. Type l3,2 . Janet basis {ξy , ηx , ηy − cξx , ξxx }. g8 : {∂y , y∂y , y 2 ∂y }. Type l3,1 . Janet basis {ξ, ηx , ηyyy }. g : {∂x + ∂y , x∂x + y∂y , x2 ∂x + y 2 ∂y }. Type l3,1 . Janet basis 10 2 2 (η − ξ), ξ − 2 ξ − ξy , ηx , ηy + ξx + x − xx y x − y x (x − y)2 (η − ξ) . g13 : {∂x , 2x∂x + y∂y , x2 ∂x + xy∂y }. Type l3,1 .  Janet basis ξx − y2 η, ξy , ηy − y1 η, ηxx . g15 (r = 3): {φ1 ∂y , φ2 ∂y , φ3 ∂y }. Type l3,6 . Janet basis {ξ, ηy , ηxxx + q1 ηxx + q2 ηx + q3 η}.

Lie Transformation Groups

141

g16 (r = 2): {φ1 ∂y , φ2 ∂y , y∂y }. Type l3,2 with c = 1. Janet basis {ξ, ηxx + q1 ηx + q2 η, ηxy , ηyy }. g17 (l = 1, ρ1 = 1, α1 = 0): {∂x , ∂y , x∂y }. Type l3,5 . Janet basis {ξx , ξy , ηy , ηxx }. g17 (l = 1, ρ1 = 1, α1 = 1): {∂x , ex ∂y , xex ∂y }. Type l3,3 . Janet basis {ξx , ξy , ηy , ηxx − 2ηx + η}. g17 (l = 2, ρ1 = ρ2 = 0, α1 = 0, α2 = 1): {∂x , ∂y , ex ∂y }. Type l3,4 . Janet basis {ξx , ξy , ηy , ηxx − ηx }. g17 (l = 2, ρ1 = ρ2 = 0, α1 = 1, α2 = c): {∂x , ex ∂y , ecx ∂y }, c 6= 0, 1. Type l3,2 . Janet basis {ξx , ξy , ηy , ηxx − (c + 1)ηx + cη}. g20 (r = 1): {∂x , ∂y , x∂x + (x + y)∂y }. Type l3,3 . Janet basis {ξy , ηx − ξx , ηy − ξx , ξxx }. g24 : {∂x , ∂y , x∂x + y∂y }. Type l3,2 with c = 1. Janet basis {ξy , ηx , ηy − ξx , ξxx }. Four-parameter groups g6 : {∂x , ∂y , x∂x , y∂y }. Type l4,10 . Janet basis {ξy , ηx , ξxx , ηyy }. g9 : {∂x , ∂y , y∂y , y 2 ∂y }. Type l4,1 . Janet basis {ξx , ξy , ηx , ηyyy }. g14 : {∂x , x∂x , x2 ∂x + xy∂y , y∂y }. Type l4,1 .  Janet basis ξy , ηy − y1 η, ξxx − y2 ηx , ηxx . g15 with r = 4: {φ1 ∂y , φ2 ∂y , φ3 ∂y , φ4 ∂y }. Type l4,17 . Janet basis {ξ, ηy , η4x + q1 ηxxx + q2 ηxx + q3 ηx + q4 η}. g16 with r = 3: {φ1 ∂y , φ2 ∂y , φ3 ∂y , y∂y }. Type l4,6 with a = b = 1. Janet basis {ξ, ηxy , ηyy , ηxxx + q1 ηxx + q2 ηx + q3 η}. g17 with l = 1, ρ1 = 2, α1 = 0: {∂x , ∂y , x∂y , x2 ∂y }. Type l4,13 . Janet basis {ξx , ξy , ηy , ηxxx }. g17 with l = 1, ρ1 = 2, α1 = 1: {∂x , ex ∂y , xex ∂y , x2 ex ∂y }. Type l4,2 . Janet basis {ξx , ξy , ηy , ηxxx − 3ηxx + 3ηx − η}. g17 with l = 2, ρ1 = 0, ρ2 = 1, α1 = 0, α2 = 1: {∂x , ∂y , ex ∂y , xex ∂y }. Type l4,8 . Janet basis {ξx , ξy , ηy , ηxxx − 2ηxx + ηx }. g17 with l = 2, ρ1 = 1, ρ2 = 0, α1 = 0, α2 = 1: {∂x , ∂y , x∂y , ex ∂y }. Type l4,11 . Janet basis {ξx , ξy , ηy , ηxxx − ηxx }. g17 with l = 2, ρ1 = 1, ρ2 = 0, α1 = 1, α2 = α: {∂x , ex ∂y , xex ∂y , eαx ∂y }. Type l4,3 . Janet basis {ξx , ξy , ηy , ηxxx − (α + 1)ηxx + (2α + 1)ηx − αη}. g17 with l = 3, ρ1 = ρ2 = ρ3 = 0, α1 = 0, α2 = 1, α3 = α: {∂x , ∂y , ex ∂y , eαx ∂y }. Type l4,9 . Janet basis {ξx , ξy , ηy , ηxxx − (α + 1)ηxx + αηx }. g17 with l = 3, ρ1 = ρ2 = ρ3 = 0, α1 = 1, α2 = α, α3 = β : {∂x , ∂y , eαx ∂y , eβx ∂y }. Type l4,6 . Janet basis {ξx , ξy , ηy , ηxxx − (α + β)ηxx + αβηx }. g18 with l = 1, ρ1 = 1, α1 = 0: {∂y , x∂y , y∂y , ∂x }. Type l4,12 . Janet basis {ξx , ηy , ξyy , ηxx }. g18 with l = 2, ρ1 = ρ2 = 0, α1 = 0, α2 = 1: {∂y , ex ∂y , y∂y , ∂x }. Type l4,10 , Janet basis {ξx , ξy , ηxx − ηx , ηxy , ηyy }. g19 with r = 2: {∂y , x∂y , ∂x , x∂x + cy∂y }. Type l4,5 (c). Janet basis {ξy , ηy − cξx , ξxx , ηxx }.

142 g20 with r = 2: {∂x , ∂y , x∂y , x∂x + (x2 + 2y)∂y }. Type l4,4 . Janet basis {ξy , ηy − 2ξx , ξxx , ηxx − 2ξx }. Selected Five-parameter groups g1 : {∂x , ∂y , x∂y , x∂x − y∂y , y∂x }. Janet basis {ηy + ξx , ξxx , ξxy , ξyy , ηxx }. g11 : {∂x , ∂y , y∂y , x∂x , y 2 ∂x }. Janet basis {ηy , ξxx , ξxy , ξyy − y1 ξy , ηxx }. g15 (r = 5): {φ1 ∂y , φ2 ∂y , φ3 ∂y , φ4 ∂y , φ5 ∂y }. Janet basis {ξ, ηy , η5x + q1 η4x + q2 ηxxx + q3 ηxx + q4 ηx + q5 η}. g16 (r = 4): {φ1 ∂y , φ2 ∂y , φ3 ∂y , φ4 ∂y , y∂y }. Janet basis {ξ, ηxy , ηyy , η4x + q1 ηxxx + q2 ηxx + q3 ηx + q4 η}. g18 (l = 1, ρ1 = 3, α1 = 0): {∂y , x∂y , x2 ∂y , y∂y , ∂x }. Janet basis {ξx , ξy , ηxy , ηyy , ηxxx }. g18 (l = 2, ρ1 = 1, ρ2 = 0, α1 = 0, α2 = 1): {∂y , x∂y , ex ∂y , y∂y , ∂x }. Janet basis {ξx , ξy , ηxy , ηyy , ηxxx − ηxx }. g18 (l = 3, ρ1 = ρ2 = ρ3 = 0, α1 = 0, α2 = 1, α3 = c), c 6= 0, 1: {∂y , x∂y , ecx ∂y , y∂y , ∂x }. Janet basis {ξx , ξy , ηxy , ηyy , ηxxx − (c + 1)ηxx + cηx }. g19 (r = 3): {∂y , x∂y , x2 ∂y , ∂x , x∂x +cy∂y }. Janet basis {ξy , ηy −cξx , ξxx , ηxxx }. g20 (r = 3): {∂y , x∂y , x2 ∂y , ∂x , x∂x + (3y + x3 )∂y }. Janet basis {ξy , ηy − 3ξx , ξxx , ηxxx − 6ξx }. g21 (r = 2): {∂x , ∂y , x∂y , y∂y , x∂x }. Janet basis {ξy , ξxx , ηxx , ηxy , ηyy }. g22 (r = 2): {∂x , ∂y , x∂y , 2x∂x + y∂y , x2 ∂x + xy∂y }. Janet basis {ξy , ηy − 12 ξx , ηxx , ξxxx }. Selected Six-parameter groups g2 : {∂x , ∂y , x∂y , y∂y , x∂x , y∂x }. Janet basis {ξxx , ξxy , ξyy , ηxx , ηxy , ηyy }. g12 : {∂x , ∂y , y∂y , x∂x , y 2 ∂y , x2 ∂x }. Janet basis {ξy , ηx , ξxxx , ηyyy }. g15 (r = 6): {φ1 ∂y , φ2 ∂y , φ3 ∂y , φ4 ∂y , φ5 ∂y , φ6 ∂y }. Janet basis {ξ, ηy , η6x + q1 η5x + q2 η4x + q3 ηxxx + q4 ηxx + q5 ηx + q6 η}. g16 (r = 5): {φ1 ∂y , φ2 ∂y , φ3 ∂y , φ4 ∂y , φ5 ∂y , y∂y }. Janet basis {ξ, ηxy , ηyy , η4x + q1 ηxxx + q2 ηxx + q3 ηx + q4 η}. g19 (r = 4): {∂y , x∂y , x2 ∂y , x3 ∂y , ∂x , x∂x + cy∂y }. Janet basis {ξy , ξxx , ηxy , ηyy , ηxxxx }. Selected Seven- and Eight-parameter groups g23 (r = 3) = {∂y , x∂y , x2 ∂y , y∂y , ∂x , x∂x , x2 ∂x + 2xy∂y }. Janet basis {ξy , ηxy − ξxx , ηyy , ξxxx , ηxxx }. g3 : {∂x , ∂y , x∂y , y∂y , x∂x , y∂x , x2 ∂x + xy∂y , xy∂x + y 2 ∂y }. Janet basis {ξyy , ηxx , ηxy , ηyy , ξxxx , ηxxy }. In addition to this classification of the finite groups of a two-dimensional manifold, Lie [110] derived a classification of its infinite groups. This listing is not given here because it is not required for the applications in this book. The only exception is group G, no. XXXV in Lie’s notation, which occurs in the symmetry analysis of first order equations.

Lie Transformation Groups

143

G: {ξ(x, y)∂x , η(y)∂y }. Janet basis {ηx }.

3.5

Lie Systems

In Chapter 2 general systems of linear homogeneous pde’s have been discussed without any assumptions on their origin or special properties of their solutions. The uniqueness of the Janet basis representation makes it possible to express additional knowledge in terms of the Janet basis coefficients. For the applications in this book, determining the symmetries of ordinary differential equations, this additional knowledge is of threefold origin.  Lie’s relations must always be true because they express the general properties of commutators of vector fields.  Geometric relations express geometric properties of particular group types, e. g. the number of systems of imprimitivity.  Algebraic relations express algebraic properties of particular Lie algebra types, e. g. in terms of the structure of its characteristic polynomial. The importance of knowing the explicit form of the coherence conditions given in Theorems 2.15, 2.16 and 2.17 becomes obvious at this point because only this knowledge, combined with the additional constraints, allows it to identify the symmetry classes of ode’s. Constraints from Lie’s Relations. Let a Janet basis with k ≥ 2 parametric derivatives be given, whose solutions are the coefficients ξ(x, y) and η(x, y) of k vector fields generating a finite group of the plane with coordinates x and y. Let (ξ1 , η1 ) and (ξ2 , η2 ) be any pair of solutions. Then according to (3.11) a third solution (ξ3 , η3 ) is given by ξ3 = ξ1 ξ2,x − ξ1,x ξ2 + η1 ξ2,y − ξ1,y η2 , η3 = ξ1 η2,x − η1,x ξ2 + η1 η2,y − η1,y η2 .

(3.28)

This property leads to severe additional restrictions for the coefficients of the corresponding Janet basis. In order to obtain them explicitly, the condition that ξ3 and η3 as defined by (3.28) be a solution would have to be expressed in terms of these coefficients. To this end at first the right hand sides of (3.28) are reduced w.r.t. the Janet basis at hand. From the resulting expressions those derivatives of ξ3 and η3 that occur in the Janet basis are calculated and are again reduced w.r.t. to the Janet basis. The result is substituted into the Janet basis. Its elements become linear and homogeneous in 12 k(k −1) bilinear expressions of the form pi qj − pj qi where the pj , qj traverse the k parametric derivatives involved. Their coefficients yield the desired constraints. Due to

144 its importance in later parts of this book, a special term for systems of pde’s obeying Lie’s relations is introduced. Definition 3.8 (Lie system). A system of linear homogeneous pde’s with the property that (ξ3 , η3 ) defined by (3.28) is a solution if this is true for (ξ1 , η1 ) and (ξ2 , η2 ), is called a Lie system. In the remaining part of this subsection the complete answer for several Janet basis types is given explicitly. Obviously this concept makes sense only for Janet bases of type J (2,2) of order not less than two. For Janet bases of order two the complete answer is given next. (2,2)

(2,2)

Theorem 3.17 For type J2,1 , . . . , J2,5 as follows. The Janet basis (2,2)

J2,1

:

Janet bases Lie’s relations are

ξ = 0, ηy + a1 ηx + a2 η = 0, ηxx + b1 ηx + b2 η = 0

(3.29)

is a Lie system if its coefficients satisfy the system of pde’s a2,x − 21 b1,y − 12 a1 b2 = 0, a1,y − a1 a2 = 0, b1,xy − 2a1,x b2 + b1,y b1 − 2b2,y − b2,x a1 + a1 b1 b2 = 0, a1,xx − a1,x b1 − b1,x a1 + a1 b2 = 0. The Janet basis (2,2)

J2,2

:

ξ = 0, ηx + a1 η = 0, ηyy + b1 ηy + b2 η = 0

(3.30)

is a Lie system if its coefficients satisfy the system of pde’s a1 = 0, b2,x = 0, b1,x = 0, b1,y − b2 = 0. The Janet basis (2,2)

J2,3

:

ξx + a1 η + a2 ξ = 0, ξy + b1 η + b2 ξ = 0, ηx + c1 η + c2 ξ = 0, ηy + d1 η + d2 ξ = 0

(3.31)

is a Lie system if its coefficients satisfy the system of pde’s d1,x − d2,y + b1 c2 − b2 c1 = 0, c1,y − d2,y + a1 d2 − b2 c1 = 0, c2,y − d2,x + a2 d2 − b2 c2 − c1 d2 + c2 d1 = 0, c1,x − d2,x + a1 c2 − a2 c1 + a2 d2 − b2 c2 = 0, b1,x − b2,y − a1 b2 + a2 b1 − b1 d2 + b2 d1 = 0, a2,y − b2,x − a1 d2 + b1 c2 = 0, a1,x − b2,x − a1 d2 + b2 c1 = 0, a1,y − b2,y − a1 d1 + b1 c1 − b1 d2 + b2 d1 = 0.

(3.32)

The Janet basis (2,2)

J2,4

:

η + a2 ξ = 0, ξy + b1 ξx + b2 ξ = 0, ξxx + c1 ξx + c2 ξ = 0

(3.33)

Lie Transformation Groups

145

is a Lie system if its coefficients satisfy the system of pde’s a2,y b1 + a2,x b21 + b1,y a2 − b1,x − a2 b1 b2 − b2 , b2,xx − 2b1,x c2 + b2,x c1 − c2,y − c2,x b1 = 0, b1,xx − b1,x c1 + 2b2,x − c1,y − c1,x b1 = 0, a2,xx + a2,x (2b1,x − b1 c1 ) − (2b2,x − c1,y )a2 − c1,x + (a2 b1 + 1)c2 = 0. The Janet basis (2,2)

J2,5

:

η + a1 ξ = 0, ξx + b1 ξ = 0, ξyy + c1 ξy + c2 ξ = 0

(3.34)

is a Lie system if its coefficients satisfy the system of pde’s b1,y − 21 c1,x = 0, a1,x − a1 b1 = 0, c1,xy + c1,x c1 − 2c2,x = 0, a1,yy − a1,y c1 − c1,y a1 + a1 c2 = 0. Proof The proof is similar for all five cases. Therefore it is given only (2,2) for type J2,3 Janet bases. Reduction of the right hand sides of (3.28) w.r.t. (3.31) where ξ and η are replaced by (ξ1 , η1 ) or (ξ2 , η2 ), yields ξ3 = (b2 − a1 )B,

η3 = (d2 − c1 )B where B ≡ ξ1 η2 − η1 ξ2

(3.35)

and the useful relations Bx = −(a2 + c1 )B, By = −(b2 + d1 )B. The reduced first order partial derivatives of ξ3 and η3 are   ξ3,x = b2,x − a1,x − (b2 − a1 )(a2 + c1 ) B,   ξ3,y = b2,y − a1,y − (b2 − a1 )(b2 + d1 ) B, (3.36)  η3,x = d2,x − c1,x − (d2 − c1 )(a2 + c1 )]B,   η3,y = d2,y − c1,y − (d2 − c1 )(b2 + d1 ) B with B as above. Substituting (3.35) and (3.36) into the Janet basis (3.31) and equating the coefficients of the parametric derivatives to zero yields the following relations for the coefficients a1 , a2 , . . . , d2 and its first derivatives. b2,x − a1,x − (b2 − a1 )c1 + (d2 − c1 )a1 = 0, b2,y − a1,y − (b2 − a1 )d1 + (d2 − c1 )b1 = 0, d2,x − c1,x + (b2 − a1 )c2 − (d2 − c1 )c1 = 0, d1,y − c1,y + (b2 − a1 )d2 − (d2 − c1 )b2 = 0. These conditions are necessary and sufficient for (3.31) to be a Lie system. They have to be supplemented by the integrability conditions of Theorem 2.17. After some final reductions have been performed, the above system (3.32) of pde’s for the coefficients is obtained.

146 Example 3.32

(2,2)

Consider the type J2,3

system

z1,x +

2(y 2 − 1) 2 1 z2 − z1 = 0, z1,y + 2 2x z2 − z1 = 0, y(y − 2) x(y 2 − 2) y (y − 2) y(y 2 − 2)

z2,x +

2(y 2 − 1) 2y 3 2y 2 2 z2 − 2 2 z1 = 0, z2,y − z2 + z1 = 0. 2 x(y − 2) x (y − 2) y(y − 2) x(y 2 − 2)

2

2

Because its coefficients satisfy the coherence conditions of Theorem 2.16 it is 2 a Janet basis
and has a two-dimensional solution space, a basis is (x , 2xy) x 2 and y , y . Because the constraints (3.32) are violated it is not a Lie system. 2 Consequently, the commutator of U1 = x2 ∂x + 2xy∂y and U2 = x y ∂x + y ∂y 2

2 which is [U1 , U2 ] = − 3x y ∂x + 2x(y − 1)∂y is not a linear combination of U1

and U2 over the constants, i.e., U1 and U2 do not generate a Lie algebra. Example 3.33

(2,2)

Consider the type J2,3

system

1 y 1 1 1 ξx − ξ = 0, ξy = 0, ηx − η+ ξ = 0, ηy − η− ξ = 0. x x+y x(x + y) x+y x+y It satisfies the constraints (3.32) that include the coherence conditions of Theorem 2.16, consequently it is a Lie system. A solution basis is (x, y) and (x, −x) from which the generators U1 = x∂x + y∂y and U2 = x(∂x − ∂y ) follow with [U1 , U2 ] = 0. Example 3.34 ξx −

(2,2)

Consider the type J2,3

system

3x2 + 1 2y 1 ξ = 0, ξy = 0, ηx + 2 ξ = 0, ηy − η = 0. y x2 + x x +1

It satisfies the constraints (3.32) that include the coherence conditions of Theorem 2.16, consequently it is a Lie system. A solution basis is (0, y) and (x3 + x, −x2 y). It is obtained from the preceding example by the variable y¯ transformation x = x ¯y¯ and y = x ¯ and replacing the barred variables by x and y. As these examples show the main advantage of applying Theorem 3.17 is that it does not require one to know the solutions; it uses only the coefficients of a given system and arithmetic operations and differentiations. For higher order systems Lie’s relations become increasingly more complicated due to the larger number of coefficients involved, therefore they cannot be given explicitly in this book. Only a single third order system will be treated in detail because it occurs frequently in later applications. Theorem 3.18

(2,2)

The type J3,4

Janet basis

ξx + a1 η + a2 ξ = 0, ξy + b1 η + b2 ξ = 0, ηy + c1 ηx + c2 η + c3 ξ = 0, ηxx + d1 ηx + d2 η + d3 ξ = 0

(3.37)

Lie Transformation Groups

147

is a Lie system if its coefficients satisfy the following system of pde’s. b2 = 0, b3 = 0, a2 c1 = 0, d2,x − d3,y + (2a2 − c2 )d3 − (2a3 − c3 )d2 = 0, c3,x c1 + a2 (a3 − 2c3 + d1 ) − c1 (c23 + c1 d3 − c3 d1 + d2 ) = 0, c3,y − 12 d1,y + 21 a2 (a3 − 3c3 + d1 ) − 21 c1 d2 = 0, c2,x − 12 d1,y + 12 a2 (a3 − 3c3 + d1 ) − 12 c1 d2 = 0, a3,x − 2c3,x + d1,x + a3 (2c3 − a3 − d1 ) + c1 d3 = 0, a3,y − a2 c3 = 0, c1,x + c1 (a3 − c3 ) = 0, c1,y − c1 c2 = 0, a2,y − a2 c2 = 0, d1,xy − 3c3,x a2 + d1,y d1 − 2d2,y − d3,y c1 − a2 a3 (a3 − 3c3 + 2d1 ) +c1 d2 (2c3 + a3 + d1 ) + a2 c23 − c1 c2 d3 − a2 d21 = 0, c3,xx + c3,x (d1 − 2c3 − 2a3 ) + d1,x c3 − d3,y − d3,x c1 +2a3 c3 (c3 − d1 ) + c1 d3 (3a3 − c3 ) − c2 d3 + c3 d2 = 0. (3.38) Proof Proceeding similarly to the preceding theorem, at first ξ3 and η3 are obtained in the form ξ3 = (a2 − b3 )B3 ,

η3 = c1 B1 − B2 − c3 B3

(3.39)

where the three bilinear expressions B1 , B2 and B3 are defined in terms of the parametric derivatives ηx , η, ξ with the respective index, and B1 = η1,x η2 − η1 η2,x , B2 = η1,x ξ2 − ξ1 η2,x , B3 = η1 ξ2 − ξ1 η2 .

(3.40)

The following relations are useful for the subsequent calculations. B1,x = −d1 B1 + d3 B3 , B1,y = −(c1,x − 2c2 − c1 d1 )B1 − c3 B2 + (c3,x − a3 c3 − c1 d3 )B3 , B2,x = −a2 B1 − (a3 + d1 )B2 − d2 B3 (3.41) B2,y = −b2 B1 − (c1,x + b3 + c2 − c1 d1 )B2 − (c2,x − a1 c3 − c1 d2 )B3 , B3,x = B2 − a3 B3 , B3,y = −c1 B2 − (b3 + c2 )B3 . Applying these expressions, the necessary derivatives of ξ3 and η3 may be computed from (3.39). If they are substituted into (3.37), the coefficients of the B 0 s yield a system of equations for the coefficients a1 , a2 , . . . , d3 . Combined with the integrability conditions of Theorem 2.17 they represent the necessary and sufficient conditions (3.38) such that the general solution of (3.37) generates a three-dimensional Lie algebra of vector fields. The system of pde’s (3.38) suggests that it may be split into several alternatives due to algebraic factorization of some of its members. It turns out that altogether three systems are generated in this way as is shown next.

148 Corollary 3.1

The system (3.38) splits into the following alternatives.

c1 = 0, b3 = 0, b2 = 0, a3 − 2c3 + d1 = 0, d2,x − d3,y + d3 (2a2 − c2 ) + d2 (2d1 − 3c3 ) = 0,

i)

c3,y − 21 d1,y − 12 a2 c3 = 0, c2,x − 21 d1,y − 12 a2 c3 = 0, a2,x − a2 c3 = 0, a2,y − a2 c2 = 0, d1,xy − 3a2 c3,x + d1,y d1 − 2d2,y + 3a2 c3 (c1 − d1 ) = 0, c3,xx − 3c3,x (2c3 − d1 ) + d1,x c3 − d3,y +2c23 (2c3 − 3d1 ) + 2c3 d21 − c2 d3 + c3 d2 = 0.

ii)

c1 = 0, b3 = 0, b2 = 0, a2 = 0, d2,x − d3,y − 2a3 d2 − c2 d3 + c3 d2 = 0, c3,y − 21 d1,y = 0, c2,x − 12 d1,y = 0, a3,x − 2c3,x + d1,x − a23 + 2a3 c3 − a3 d1 = 0, a3,y = 0, d1,xy + d1 d1,y − 2d2,y = 0, c3,xx − c3,x (2a3 + 2c3 − d1 ) + d1,x c3 − d3,y +2a3 c3 (c3 − d1 ) − c2 d3 + c3 d2 = 0.

iii)

b3 = 0, b2 = 0, a2 = 0, d2,x − d3,y − 2a3 d2 − c2 d3 + c3 d2 = 0, c3,x − c1 d3 − c23 + c3 d1 − d2 = 0, c3,y − 21 d1,y − 12 c1 d2 = 0, c2,x − 21 d1,y − 12 c1 d2 = 0, c1,x + c1 (a3 − c3 ) = 0, c1,y − c1 c2 = 0, a3,x + d1,x + (a3 + d1 )(2c3 − a3 ) = 0, a3,y = 0,

d1,xy + d1 d1,y − 2d2,y − d3,y c1 + c1 d2 (a3 + d1 − 2c3 ) − c1 c2 d3 = 0. The next example shows how additional information beyond Lie’s conditions is necessary in order to distinguish certain group types. (2,2)

Example 3.35 Three type J3,4 coefficients of (3.37) as follows. 2, d = 2, a) a3 = − x 1 x

Lie systems are defined by nonvanishing

2, d = 2 + 2 , d = 1 , b) a3 = − x 1 x x2 2 x4 2 2 1 c) a3 = − x , d1 = x + 2 . x All three cases satisfy system (3.38). Case a) satisfies only alternative i) of Corollary 3.1, case b) satisfies only alternative ii) whereas case c) satisfies alternatives ii) and iii).

Lie Transformation Groups

149

Geometric Constraints. In the symmetry analysis of ode’s the most important distinguishing feature for the type of a symmetry group which is of geometric origin is its system of imprimitivity. Therefore for those Janet basis types that occur in the symmetry analysis the constraints determining their systems of imprimitivity are described in this subsection. They are based on Theorem 3.16. For the two first order bases the answer is rather simple. Theorem 3.19

The Lie systems of types (2,2)

J1,1

: ξ = 0, ηx + aη = 0, ηy + bη = 0,

(2,2)

: η + aξ = 0, ξx + bξ = 0, ξy + cξ = 0

J1,2

define one-parameter groups allowing a system of imprimitivity depending on an undetermined function. Proof

The systems of imprimitivity are determined by

ωy + bω + a = 0 or ωx + aωy + cω 2 + (ay − b − c)ω + ax − ab = 0 respectively. The general solution of either of these pde’s for ω(x, y) contains an undetermined function of a single argument. Consequently, they determine systems of imprimitivity depending on a function. This result was to be expected because one-parameter groups according to the tabulation on page 139 always allow a system of imprimitivity of this kind. (2,2)

(2,2)

Theorem 3.20 The Lie systems of types J2,1 , . . . , J2,5 define vector fields for two-parameter groups with the following systems of imprimitivity. (2,2)

J2,1 : System (3.29) does not allow a one-parameter system of imprimitivity. It allows two systems of imprimitivity if a1 6= 0, and a single one if a1 = 0. (2,2)

J2,2 : System (3.30) always allows two systems of imprimitivity. (2,2)

J2,3 : System (3.31) always defines a one-parameter system of imprimitivity. (2,2)

J2,4 : System (3.33) always allows two systems of imprimitivity. (2,2)

J2,5 : System (3.34) does not allow a one-parameter system of imprimitivity. It allows two systems of imprimitivity if a1 6= 0 and a single one if a1 = 0. (2,2)

Proof Type J2,1 : There is always the system of imprimitivity corresponding to ξy = 0. Any other system of imprimitivity is determined by the equations a1 ω − 1 = 0, ωy + a2 ω = 0. If a1 6= 0 there is one solution, if a1 = 0 the first equation becomes inconsistent, consequently a solution does not exist.

150 (2,2)

Type J2,2 : There are always the systems of imprimitivity corresponding to ξy = 0, and a second one that follows from the equations ω = 0, ωy +a1 = 0 with the constraint a1 = 0 according to Theorem 3.17. (2,2) Type J2,3 : Any system of imprimitivity is determined by the equations ωx − b2 ω 2 − (a2 − d2 )ω + c2 = 0, ωy − b1 ω 2 − (a1 − d1 )ω + c1 = 0. Due to Theorem 3.17 it satisfies the coherence conditions (2.51). Consequently, it determines always a system of imprimitivity depending on a parameter. (2,2) Type J2,4 : Any system of imprimitivity is determined by the equations b1 ω 2 + (a2 b1 − 1)ω = 0, a2 ωy − ωx + b2 ω 2 − (a2,y − a2 b2 )ω − a2,x = 0. If b1 6= 0, the first equation determines two systems of imprimitivity because a1 b1 + 1 6= 0. If b1 = 0, by Theorem 3.17 it follows b2 = 0, i.e., there is one system of imprimitivity corresponding to ξy = 0, and a second one follows from ω = 0. (2,2) Type J2,5 : Any system of imprimitivity is determined by the equations ω(ω + a1 ) = 0, a1 ωy − ωx − (a1,y + b1 )ω − a1,x + a1 b1 = 0. There is always the solution ω = 0. If a1 6= 0 there is a second system of imprimitivity corresponding to ω = −a1 . There are three Janet basis types that occur in the symmetry analysis of second and third order ode’s in Chapter 5, their systems of imprimitivity are considered next. (2,2)

(2,2)

(2,2)

Theorem 3.21 The Lie systems of types J3,4 , J3,6 and J3,7 define vector fields for three-parameter groups with the following systems of imprimitivity. (2,2)

J3,4 : The Lie system ξx + a2 η + a3 ξ = 0, ξy + b2 η + b3 ξ = 0, ηy + c1 ηx + c2 η + c3 ξ = 0, ηxx + d1 ηx + d2 η + d3 ξ = 0

(3.42)

defines a three-parameter group allowing two systems of imprimitivity if c1 6= 0 and a single one if c1 = 0. There cannot be a system of imprimitivity depending on a parameter. (2,2)

J3,6 : The Lie system ξy + a1 ξx + a2 η + a3 ξ = 0, ηx + b1 ξx + b2 η + b3 ξ = 0, ηy + c1 ξx + c2 η + c3 ξ = 0, ξxx + d1 ξx + d2 η + d3 ξ = 0

(3.43)

Lie Transformation Groups

151

defines a three-parameter group allowing a system of imprimitivity depending on a parameter if a1 = b1 = 0 and c1 = −1, two systems of imprimitivity if either ai = bi = 0, i = 1, 2, 3 or D ≡ 4a1 b1 + (c1 + 1)2 6= 0, and a single one if D = 0 and a1 6= 0 or b1 6= 0 or c1 6= −1. (2,2)

J3,7 : The Lie system ξx + a2 η + a3 ξ = 0, ηx + b1 ξy + b2 η + b3 ξ = 0, ηy + c1 ξy + c2 η + c3 ξ = 0, ξyy + d1 ξy + d2 η + d3 ξ = 0

(3.44)

defines a two-parameter group which does not allow a system of imprimitivity depending on a parameter. It allows two systems of imprimitivity if D ≡ b1 − 41 c21 6= 0 and a single one if D = 0. Proof

(2,2)

Type J3,4 : According to Theorem 3.18, for any Janet basis (2,2) J3,4

of type there holds b2 = b3 = 0, i.e., there is always a system of imprimitivity corresponding to ξy = 0. A possible second one is determined by the equations c1 ω − 1 = 0, ωx + (c3 − a3 )ω = 0, ωy + (c2 − a2 )ω = 0. The coherence conditions for the system of first order pde’s are satisfied due to Theorem 3.18. Consequently, it allows an additional system of imprimitivity if and only if c1 6= 0. (2,2) J3,6 : Any system of imprimitivity is determined by the equations a1 ω 2 −(c1 +1)ω −b1 = 0, ωx −a3 ω 2 +c3 ω +b3 = 0, ωy −a2 ω 2 +c2 ω +b2 = 0. If the first equation vanishes identically the remaining system of pde’s which is coherent defines a system of imprimitivity depending on a parameter. If not, the discriminant of the first equation determines the number of different solutions and thereby the systems of imprimitivity. If the first equation reduces to ξy = 0 or the second one to ηx = 0, an additional system of imprimitivity is obtained. (2,2) J3,7 : Any system of imprimitivity is determined by the equations ω 2 + c1 ω + b1 = 0, ωx + (c3 − a3 )ω + b3 = 0, ωy + (c2 − a2 )ω + b2 = 0. Because the quadratic equation for ω cannot vanish identically and the system of pde’s for ω is coherent, the discriminant D of the quadratic equation determines the number of imprimitivity systems. Example 3.36 For all three alternatives of Corollary 3.1 there holds b2 = b3 = 0, i.e., by the above lemma there is always one system of imprimitivity corresponding to ξy = 0. Reducing condition (3.26) by (3.42) and

152 separating the result w.r.t. the parametric derivatives leads to the system c1 ω − 1 = 0, ωy − b2 ω 2 + (c2 − a2 )ω = 0, ωx − b3 ω 2 + (c3 − a3 )ω = 0. If c1 = 0 the first equation becomes inconsistent and a second system of imprimitivity does not exist. If c1 6= 0, ω = c11 is the unique solution of the full system and generates a second system of imprimitivity. Summing up this means, alternatives i) and ii) allow a single system of imprimitivity, alternative iii) allows two of them. There occur two Lie systems of order four which have to be analyzed w.r.t. to their group properties. Their systems of imprimitivity are considered next. (2,2)

(2,2)

Theorem 3.22 The Lie systems of types J4,9 and J4,14 define vector fields for four-parameter groups with the following systems of imprimitivity. (2,2)

J4,9 : The Lie system ξy + a2 ξx + a3 η + a4 ξ = 0, ηy + b1 ηx + b2 ξx + b3 η + b4 ξ = 0, ξxx + c1 ηx + c2 ξx + c3 η + c4 ξ = 0, ηxx + d1 ηx + d2 ξx + d3 η + d4 ξ = 0 (3.45) defines a four-parameter group which does not allow a system of imprimitivity depending on a parameter. It allows two systems of imprimitivity if a1 = 0, a2 = a3 = a4 = 0, b2 + 1 = 0 and b1 6= 0, and a single one if a2 = a3 = a4 = b1 = 0. (2,2)

J4,14 : The Lie system ηx + a1 ξy + a2 ξx + a3 η + a4 ξ = 0, ηy + b1 ξy + b2 ξx + b3 η + b4 ξ = 0, ξxx + c1 ξy + c2 ξx + c3 η + c4 ξ = 0, ξxy + d1 ξy + d2 ξx + d3 η + d4 ξ = 0, ξyy + e1 ξy + e2 ξx + e3 η + e4 ξ = 0 (3.46) defines a four-parameter group which does not allow a system of imprimitivity depending on a parameter. It allows two systems of imprimitivity if a1 = 0, a2 = a3 = a4 = 0, b2 + 1 = 0 and b1 6= 0 or a2 = 0, b2 + 1 = 0, b21 − 4a1 6= 0 and a1 6= 0 or a3 6= 0 or a4 6= 0. The proof is similar as for the preceding theorems and is therefore skipped. Algebraic Relations. The third set of constraints for the coefficients of Janet bases occurring in the symmetry analysis of ode’s originates from the algebraic properties of their Lie algebras. They are based on Theorem 3.7, in particular on part ii) describing the properties of characteristic polynomials. The Lie algebra types li,j have been defined in Section 3.4 on page 135 ff. (2,2)

Theorem 3.23 The Lie system (3.31) of type J2,3 algebra of vector fields of the following types:

may define a Lie

l2,1 if a1 6= b2 or c1 6= d2 . l2,2 if a1 = b2 and c1 = d2 .

Lie Transformation Groups

153

(2,2)

Proof For the general Lie system of type J2,3 the characteristic polynomial has the form ω 2 − [(c1 − d2 )E2 − (a1 − b2 )E1 ]ω. The above coefficient constraints generate the structure of the two-dimensional Lie algebras l2,1 or l2,2 respectively listed on page 135. (2,2)

Theorem 3.24 The Lie system (3.42) of type J3,4 algebra of vector fields of the following types:

may define a Lie

l3,1 if c1 = 0, a3 − 2c3 + d1 = 0 and either a2 6= 0 or c3,x − c23 + c3 d1 − d2 6= 0. l3,2 (c) if c3,x + 41 a3 (a3 − 4c3 + 2d1 ) + 14 d21 − d2 6= 0, c 6= 0, c2 +

2c3,x + 2c23 − 2c3 d1 − 4a3 c3 + a21 + d21 − 2d2 c + 1 = 0. c3,x − c23 + c3 d1 − d2

l3,3 if a1 = 0, a2 = 0, c3,x + 41 a3 (a3 − 4c3 + 2d1 + 41 d21 − d2 = 0, a3 − 2c3 + d1 6= 0.

l3,4 if a2 = 0, c3,x − c23 + c3 d1 − c1 d3 − d2 = 0 and either c1 6= 0 or a3 − 2c3 + d1 6= 0. l3,5 if c1 = 0, a2 = 0, a3 − 2c3 + d1 = 0, c3,x − c23 + c3 d1 − c1 d3 − d2 = 0. l3,6 not possible.

Proof According to the definition (3.17) the characteristic polynomial for any three-dimensional Lie algebra has the form ω 3 + C2 ω 2 + C1 ω. In terms (2,2) of a Lie system of type J3,4 the coefficients may be expressed as follows. C2 = c1 E1 + (a3 − c3 + d1 )c1 E2 − (a3 − 2c3 + d1 )E3 , C1 = a2 c1 E1 E2 − a2 (a2 − a3 c1 + c1 c3 − c1 d1 )E22 − 2a2 E1 E3   + c3,x − c1 d3 − c23 + c3 d1 − d2 )c1 − a2 (a3 + d1 ) E2 E3 −(c3,x − c1 d3 − c23 + c3 d1 − d2 )E32 . Imposing the constraints on C1 and C2 that follow from the structure given in the listing on page 135, the above conditions are obtained after some simplifications. The condition for l3,6 follows from the fact that some structure constants are numbers different from zero.

154 Example 3.37 The three alternatives obtained in Corollary 3.1 yield the characteristic equations   ω 3 − (c3,x − c23 + c3 d1 − d2 )E32 + 2a2 c3 E3 E2 + 2a2 E3 E1 + a22 E22 ω = 0, ω 3 − (a3 − 2c3 + d1 )E3 ω 2 − (c3,x − c23 + c3 d1 − d2 )E32 ω = 0,   ω 3 − (a3 − 2c3 + d1 )E3 − (a3 c1 − c1 c3 + c1 d1 )E2 − c1 E1 ω 2 = 0 respectively. The first alternative does not contain a quadratic term but a linear one, therefore generically it corresponds to a type l3,1 Lie algebra. A type l3,2 (c = −1) Lie algebra is excluded because the coefficient of ω is not a complete square of a linear form in the E 0 s. If c3,x − c23 + c2 d1 − d2 = 0 and a2 = 0, a type l3,5 or type l3,6 Lie algebra is possible. For the second alternative, define C1 ≡ c3,x −c23 +c3 d1 −d2 , C2 ≡ a3 −2c3 +d1 and the discriminant D ≡ C22 − 4C1 . The coefficient of the first order term is proportional to a complete square of a linear form in the E 0 s. As a consequence type l3,1 is excluded. If C1 6= 0 and C2 6= 0, the condition D 6= 0 corresponds to type l3,2 (c 6= −1) and D = 0 to type l3,3 . Finally C1 = 0, C2 6= 0 corresponds to the type l3,4 , and C1 = C2 = 0 to types l3,5 or l3,6 . Due to the constraint c1 6= 0 for the last alternative, there is always a quadratic term and no linear term. Consequently it corresponds uniquely to a Lie algebra of type l3,4 . There are four additional Janet basis types the geometric properties of which have been determined in Theorems 3.21, 3.22 and 3.22. Their Lie algebra types are described in the following two theorems without proof. Example 3.38 The Lie algebra types corresponding to the three Lie systems considered in Example 3.35 are easiliy identified by the above theorem as follows: a) type l3,5 , b) type l3,3 and c) type l3,4 . (2,2)

Theorem 3.25 A Lie system (3.43) of type J3,6 of vector fields of the following types:

may define a Lie algebra

l3,1 if c1 = 1, a3 b1 − a1 b3 + b2 − c3 + d1 = 0 and either a1 b1 + 1 6= 0 or a1 (b1 − c3 ) + a3 6= 0 or a3 b1 − b2 + c3 6= 0 or a3,x − a3 (a1 b3 + c3 − d1 ) − a1 d3 − d2 6= 0 or a3,x b1 + (a1 b3 + c3 − d1 )(c3 − b2 ) − b1 d2 + d3 6= 0 or a3,x + 21 (a1 b2 − a1 c3 − a3 )(a1 b3 + c3 − d1 ) 6= 0. l3,2 (c) if c2 +

2a1 b1 + c21 + 1 c + 1 = 0, c1 6= 1. a2 b1 + c1

Lie Transformation Groups

155

l3,3 if any of the following set of conditions is satisfied. c1 + 1 = 0, b1 = 0, a1 b3 + b2 − c3 − d1 = 0, a3,x + a1 (b2 − c3 )2 − a3 b2 − a1 d3 − d2 = 0 or c1 + 1 = 0, b1 = 0, a3,x b1 − 14 a23 b21 − 14 (b2 − c1 − d1 )2 − 21 a3 b1 (b2 + c3 − d1 ) − b1 d1 = 0 or 4a1 b1 + (c1 + 1)2 = 0, b1 6= 1, (c1 + 3)(b2 − c3 ) + (c1 − 1)(d1 − a1 b3 ) + (c1 − 5)a1 b1 = 0, a3 (c1 + 1)2 (c1 − 5) + 4a1 (a1 b3 − d1 )(c1 − 1) − 4a1 (c1 + 3)(b2 − c3 ) = 0, a3,x b1 − 81 (a3 b3 − 2d3 )(c1 + 1)2 − 14 (b2 − d1 )2 − 14 (a21 b23 + a23 b21 ) − b1 d2 − 12 a3 b1 (b2 + c3 + d1 ) − 12 a1 b3 (b2 − c3 − d1 ) + 41 c3 (2b2 − c3 − 2d1 ) = 0. l3,4 if any of the following set of conditions is satisfied. a1 = 0, c1 = 0, a3 b1 = b2 − c3 , a3,x = a3 (c3 − d1 ) + d2 , c3 − b2 6= 21 d1 or a1 b1 + 1 = 0, a3 c1 + a1 (b2 − c3 ) = 0, a3 b1 = b2 − c3 , a3,x − a3 (a1 b3 + c3 − d1 ) − a1 d3 − d2 = 0, a1 b3 − 2(b2 − c3 ) 6= d1 , a3 + a1 (a1 b3 + c3 − b2 − d1 ) 6= 0. l3,5 if c1 = 1, a1 b1 + 1 = 0, a1 b3 − 2(b2 − c3 ) − d1 = 0, a3 + a1 (b2 − c3 ) = 0, (b2,x − c3,x )a1 + (b2 − c3 )(a1 d1 − a3 c1 ) + a1 d3 + d2 = 0. l3,6 not possible. (2,2)

Theorem 3.26 A Lie system (3.44) of type J3,7 of vector fields of the following types:

may define a Lie algebra

l3,1 if c1 = 0, a2 − 21 (c2 + d1 ) = 0 and either b1 6= 0 or c3 6= 0 or c2,y + d1,y − 12 c22 + 12 d21 − 2d3 6= 0. l3,2 (c) if c2 +

2b1 − c21 c + 1 = 0, c1 6= 1. b1

l3,3 if b1 − 14 c21 = 0, c3 − 12 c1 (a2 + 2c2 + 2d1 ) = 0, c2,y + d1,y + c1 d2 − 21 c22 + 12 d21 − 2d3 = 0. l3,4 if c2,y + d1,y − 2a2 (a2 + c2 + d1 ) + c1 d2 − c2 d1 − c22 − 2d3 = 0, b1 = 0, c3 − a2 c1 = 0 and either c1 6= 0 or a2 − 12 (c2 + d1 ) 6= 0. l3,5 if b1 = 0, c1 = 0, c3 = 0, a2 − 21 (c2 + d1 ) = 0, c2,y + d1,y − 12 c22 + 12 d21 − 2d3 = 0. l3,6 not possible. (2,2)

Theorem 3.27 A Lie system (3.45) of type J4,9 of vector fields of the following types:

may define a Lie algebra

156 l4,1 if a2 = b1 , b2 = 0, b1 6= 0. l4,4 if b2 = −2, b1 = a2 = 0, a4 = 21 c1 . l4,5 (c) if b2 6= 0, b2 + 1 6= 0. Then c = −b2 . (2,2)

Theorem 3.28 A Lie system (3.46) of type J4,14 may define a Lie algebra of vector fields of the following types: l4,4 if b2 = − 12 , l4,5 (c) if b2 = − 1c , c 6= 0, c 6= −2. The proofs of the preceding three theorems are similar to those of Theorems 3.23 and 3.24 and are therefore omitted. The theorems of this subsection provide the interface between geometric and algebraic properties of Lie transformation groups of the plane on the one hand, and the symmetries of ode’s expressed in terms of the Janet bases of its determining systems on the other. They will be applied frequently later in this book. It turns out that not only the Lie algebra type of the vector fields defined by a given Lie system may be obtained from its coefficients, but even its structure constants ckij w.r.t. some basis. The subsequent discussion is based on articles by Neumer [135] and Reid et al. [152]. Assume a Lie system for two functions ξ(x, y) and η(x, y) is given as a Janet basis. Let this system be of finite rank r, i.e., its general solution determines the infinitesimal generators of a group with r parameters. A special solution of this system is uniquely determined by assigning fixed values to the parametric derivatives p1 , . . . , pr . Let two special solutions ξ1 , η1 and ξ2 , η2 be determined by p1,1 , . . . , p1,r and p2,1 , . . . , p2,r respectively. The Lie system property assures that there exists a third solution ξ3 , η3 determined by p3,1 , . . . , p3,r corresponding to the commutator of the vector fields ξ1 ∂x + η1 ∂y and ξ2 ∂x + η2 ∂y and is explicitly given by (3.28). Its parametric derivatives may be computed by deriving the right hand sides of (3.28) and reducing them w.r.t. to the given Lie system. Due to the linearity of all operations involved, they are alternating bilinear forms in the parametric derivatives p1,1 , . . . , p1,r and p2,1 , . . . , p2,r and may be expressed in terms of Bk = p1,i p2,j − p1,j p2,i for k = 1, . . . , 12 r(r − 1), i, j = 1, . . . , r, i < j. This proceeding is illustrated by a three-parameter group. Example 3.39 Consider again the Janet basis {ξx , ηx + ξy , ηy , ξyy } of the preceding example. The parametric derivatives ξy , η and ξ lead to B1 = ξ1,y η2 − η1 ξ2,y , B2 = ξ1,y ξ2 − ξ2,y ξ1 , B3 = η1 ξ2 − ξ1 η2 . Differentiating and reducing the right hand sides yields ξ3,y = 0, η3 = −B2 , ξ3 = B1 , i.e., the only nonvanishing structure constants are c132 = −1 and c233 = 1. The same answer is of course obtained if they are computed from the generators ∂x , ∂y and y∂x − x∂y .

Lie Transformation Groups

157

Exercises Exercise 3.1 Show that the transformations x ¯ = x + y + a, y¯ = x − y + b do not define a group. Which part of the First Fundamental Theorem is violated? (Lie [112], vol. I, page 72). Exercise 3.2 Determine the finite transformations of the four-parameter group generated by U1 = ∂x , U2 = x∂x , U3 = y∂y , U4 = x2 ∂x + xy∂y . Exercise 3.3

Determine the order of transitivity of the group g2 .

Exercise 3.4 Show that any group of F1 (x)∂y , . . . , Fr (x)∂y with r ≥ 3 of the type g15 is similar to a group of the form ∂v , u∂v , G3 (u)∂v , . . . , Gr (u)∂v . Exercise 3.5 If the Lie algebra type of the vector fields in Example 3.1 is not a priori known, one may start with some other algebra of the listing on page 135, e. g. with type l3,1 . How do the calculations proceed in this case? Exercise 3.6 Applying (3.17) determine the characteristic polynomial for r = 2 and r = 3 explicitly. Exercise 3.7 Describe the groups g17 for l = 1, 2 or 3 in detail, i.e., list all possible groups, its generators and put up its commutator table. Exercise 3.8 For the three-parameter groups listed above determine the assignment of vector fields to the generators U1 , U2 and U3 such that the commutation relations of the respective Lie algebra have the canonical form as given in the tabulation on page 135. Exercise 3.9

The same problem for the four-parameter groups.

Exercise 3.10 The special structure of the characteristic equation entails algebraic relations for its coefficients in expanded form that are invariant w.r.t. basis transformations of the Lie algebra. Determine these relations for the algebras l4,4 and l4,5 . Exercise 3.11 Determine the assignment of vector fields of gl2 and sl2 to the generators Uk such that the commutation relations of the respective Lie algebra l4,1 and l3,1 in canonical form are obeyed.

Chapter 4 Equivalence and Invariants of Differential Equations

The two main topics of this chapter, equivalence and invariance of differential equations w.r.t. certain transformation groups, have their origin in the theory of algebraic forms as developed in the 19th century. Comprehensive introductions to the latter may be found in one of the classical text books like Clebsch [30] or Gordan [55], or in the expository article by Kung and Rota [97]. There was a general belief around the middle of the 19th century that many concepts in algebra must have an important meaning for differential equations if they were appropriately generalized. Following these ideas Cockle [31] introduced a notion that came fairly close to an invariant of a differential equation in terms of his so-called criticoids. He obtained them for linear second order ode’s by elementary methods without applying the notion of a group of transformations. Laguerre [98] was the first to realize the close connection between the type of transformations admitted by a linear second order equation and the invariants belonging to it. Brioschi [19] and Halphen [65] generalized these results to equations of third and fourth order. For quasilinear equations of first and second order Liouville [117, 120] determined various classes of invariants and distinguished absolute and relative invariants. Later, Tresse [181] in his Preisschrift presented an extensive discussion of the invariants of the equation y 00 = F (x, y, y 0 ) and solved the corresponding equivalence problem. Some of his results have been generalized to third order equations of the form y 000 = F (x, y, y 0 , y 00 ) by Leja [103]. A survey on the history of the subject, including some comments and reprints of relevant articles may be found in the booklet by Czichowski and Fritzsche [36]. A more modern treatement of the subject is given in the book by Olver [141]. In the context of differential equations the diffeomorphisms between coordinates are usually called point transformations in order to distinguish them from so-called contact transformations. The latter involve certain transformations of the first derivatives in addition to the variables themselves. This terminology will frequently be applied in the remaining part of this book. Whenever dealing with differential equations, it is essential to know whether two given equations may be transformed into each other by a suitable variable change. A special name for this fundamental concept is introduced next.

159

160 Definition 4.1 (Equivalence problem). Given two differential equations, decide whether there exists a transformation which takes one of them into the other. If the answer is affirmative the two equations are said to be equivalent under the type of transformations admitted. This definition creates a partition of all ode’s into equivalence classes. Solving an individual equation may be generalized to solving equations of a full equivalence class. To this end, in the first place it must be possible to decide membership in any given equivalence class algorithmically. As shown below, the invariants of a differential equation are a useful device for answering this question. Secondly, within a given equivalence class, a canonical representative is defined which may be applied to characterize this class. An important subproblem of this latter step is to identify the smallest function field in which the transformation functions are contained. In Chapter 5 it will be seen how the symmetry type of an equation narrows down the possible equivalence classes to which it may belong. In order to make the invariants of a differential equation into a meaningful concept, it is necessary to identify those transformations leading to a welldefined behavior of its coefficients. For linear equations this has been noticed first by St¨ ackel [173] who obtained the complete answer in this case. For a general equation Goursat [56] gave a definition of the invariants based on the behavior of its coefficients under certain groups of transformations. The subsequent discussion follows closely this latter reference. Let an n−th order ode ω(x, y, y 0 , . . . , y (n) ) = 0 be given with independent variable x and dependent variable y. Assume that it is polynomial in the derivatives, with coefficients a1 , a2 , . . . , aN depending on x and y. Let a transformation x = φ(¯ x, y¯; θ) and y = ψ(¯ x, y¯; θ) of a certain group be applied to it where θ denotes collectively the parameters or functions specifying the group element. They are limited by the requirement that the structure of the differential equation remains unchanged under its action. The structure of a differential equation may be expressed in terms of certain restrictions on its order and the power to which the various derivatives may occur. A linear equation, for example, is determined by the requirement that the dependent variable and its derivatives occur only to the first power in each term. For the largest group leaving the structure unchanged a proper name is introduced by the following definition. Definition 4.2 (Structure invariance group). Let a class of differential equations be defined in terms of certain restrictions on its structure. The largest group of transformations leaving this structure unchanged is called the structure invariance group of this class. In general it will be an infinite dimensional Lie group. Originally the concept of a structure invariance group goes back to Ovsiannikov [144], Chapter II, Section 4; he called them equivalence transformations. The name structure invariance group, however, appears to be more suggestive

Equivalence and Invariants

161

of the basic idea behind it. By definition the elements of the structure invariance group operate only on the unspecified elements of an equation, i.e., the parameters and undetermined functions that it may contain. If an equation does not contain any of them, its structure invariance group specializes to its symmetry group to be explained in the next chapter. If the structure remains unchanged under a certain transformation, it generates a new differential equation of the form ω(¯ x, y¯, y¯0 , . . . , y¯(n) ) = 0 with coefficients a ¯k = fk (a1 , . . . , aN , a1,x , a1,y , . . . , aN,x , aN,y , . . . , θ) for k = 1, . . . , N where the functions fk are uniquely determined by the transformation corresponding to θ. Consequently, it generates a group of transformations among the coefficients a1 , . . . , aN that may be prolonged to its derivatives. According to Lie [112], vol. I, pages 212 and 523, any such group allows an infinite series of differential invariants. Definition 4.3 (Invariants of an ode) Let ω(x, y, y 0 , . . . , y (n) ) = 0 be an ode which is polynomial in y, y 0 , . . . , y (n) with coefficients a1 , . . . , aN depending on x and y. If a transformation of the structure invariance group of ω with group parameters θ is applied, a new equation with transformed coefficients a ¯1 , . . . , a ¯N is obtained. An expression Φ that is transformed like Φ(¯ a1 , a ¯2 , . . . , a ¯N ) = w(θ) · Φ(a1 , a2 , . . . , aN ). is called an invariant of the ode ω with weight w. More precisely, if w is equal to 1, Φ is an absolute invariant or simply invariant, otherwise it is a relative invariant with weight w. The meaning of this relation is as follows. If the left hand side Φ is evaluated with arguments ak , an expression depending on x and y is obtained. If at the right hand side the same function Φ is evaluated with a ¯k = fk , the same expession is obtained, possibly multiplied with a factor depending only on the group parameters θ. Sometimes also a dependence on x and y is allowed. Subsequently the term invariant of a differential equation without further specification means always an invariant w.r.t. the structure invariance group of the equation under consideration. These invariants will usually be denoted by Φ. Any other invariants are denoted by small Greek letters φ, ψ, etc. The subgroup to which they correspond will be specified explicitly, or it will be clear from the context. Phrases like semi-invariants will not be applied. The problem is now to develop a set of criteria for deciding equivalence for given pairs of differential equations. It turns out that an answer may only be obtained for special classes of equations, e. g. linear equations or special types of first or second order equations. Moreover, if two equations have been shown to be equivalent, it is still an additional task to determine a transformation between them. It turns out that the necessary computations quickly lead to unmanageable equations. Therefore the following observation is vital in many

162 cases in order to replace a given problem by an equivalent one which is much easier to handle. The complexity of dealing with any class of ode’s is determined above all by the number of undetermined elements, and by the function field in which they are contained. This suggests using the undetermined elements in the admitted transformations to eliminate as many coefficients in the equation as possible, without enlarging the function field determined by the initially given equation. For the result of this transformation a special term is introduced in the following definition. Definition 4.4 (Rational normal form) A rational normal form of an ordinary differential equation that is polynomial in the dependent variable and its derivatives is an equation with the minimal number of nonconstant coefficients that may be obtained from it by a point transformation in the dependent and the independent variable. In general, a rational normal form is not unique, but defines a new structure with a structure invariance group of its own. Applying these concepts, a fairly complete discussion of linear equations is given in Section 4.1. Nonlinear equations of first order are the subject of Section 4.2. In the last Section 4.3 quasilinear equations of second order are considered.

4.1

Linear Equations

As mentioned in the introduction to this chapter, linear ode’s were historically the first class of differential equations for which a systematic study of the invariants corresponding to a group of transformations had been made in analogy to algebraic invariants. As a consequence the results are more complete for linear equations than for nonlinear ones. As usual, the general linear homogeneous equation will be written in the form y (n) + q1 y (n−1) + . . . + qn−1 y 0 + qn y = 0.

(4.1)

Structure Invariance, Normal Forms and Invariants. The subsequent theorem is originally due to St¨ackel [173]. The proof follows essentially an article of Neumer [137]. Theorem 4.1 (St¨ ackel 1893, Neumer 1937) The structure invariance group of a linear homogenous equation (4.1) consists of transformations of the form x = F (u), y = G(u)v + L(x) with F and G undetermined functions and L the general solution. Proof Let x = φ(u, v) and y = ψ(u, v) be a general point transformation. An undetermined dependence on v in the coefficients of the transformed

Equivalence and Invariants

163

equation requires φv = 0, i.e., x = F (u). Under this constraint the expression (B.9) for y (k) shows that an unmatched term proportional to v 0 v (k−1) can only be avoided if ψvv = 0, i.e., ψ(u, v) = G(u)v + H(u). Furthermore, the expression (B.9) shows that the homogeneity requires H = 0 in order to avoid any term not proportional to v or any derivative of it. L(x) is a linear combination of a fundamental system, therefore its contribution vanishes. Although the term L(x) leaves the equation invariant and therefore may be considered as trivial, it is important to recognize its existence because in some applications the maximal number of parameters involved in the structure invariance group is an important invariant. For later reference the following special case due to Neumer is formulated as a corollary. Corollary 4.1 (Neumer 1937) The structure invariance group of a linear homogeneous equation with constant coefficients is x = αu, y = eβu v + L(x) with α and β constants. Theorem 4.2

Equation (4.1) has the rational normal form y¯(n) + q2 y¯(n−2) + . . . + qn−1 y¯0 + qn y¯ = 0.

It is obtained from (4.1) by the transformations  1Z  0(n−1)/2 x = F (¯ x), y = CF exp − q1 (F )F 0 d¯ x y¯ n

(4.2)

(4.3)

where C is a constant and F (¯ x) an undetermined function. The structure invariance group of (4.2) is x ¯ = F (u),

y¯ = CF 0 (u)(n−1)/2 v + L(u)

(4.4)

where C is a constant and L is a linear combination of a fundamental system. Proof Transforming equation (4.1) according to its structure invariance group by x = F (¯ x), y = G(¯ x)¯ y leads to a new equation (see equation (B.16) of Appendix B) with the coefficient of y (n−1) equal to n

G  G0 n − 1 F 00  G − +q1 0n−1 . 0 0n F G n F F

It vanishes if G = CF

0(n−1)/2

exp



1 − n

Z

q1 F 0 d¯ x



with C a constant. This yields (4.3). Any transformation of the structure invariance group of (4.2) must have the form x ¯ = F (u), y¯ = G(u)v. Applying it to (4.2) yields an equation for v containing the term n

G  G0 n − 1 F 00  (n−1) − v . F 0n G 2 F0

(4.5)

164 It vanishes if G is proportional to F 0(n−1)/2 . Any contribution originating from L obviously drops in the final result. This yields (4.4). The transformation (4.3) is the most general one leading to rational normal form. Different choices of F (¯ x) lead to rational normal forms that are related by a transformation of its structure invariance group (4.4). Often the special choice x = x ¯ is made with the corresponding transformation 1 R q (x)dx)¯ y as in Section 2.1. y = exp (− n 1 If the coefficients of the transformed equations are allowed to be in an extension of the base field, a more special canonical form, the so-called LaguerreForsyth canonical form (Laguerre [99], Forsyth [47]), of equation (4.1) may be obtained. Theorem 4.3 (Laguerre 1879, Forsyth 1888) Any linear homogeneous equation in rational normal form y (n) + q2 y (n−2) + . . . + qn−1 y 0 + qn y = 0

(4.6)

is equivalent to an equation in Laguerre-Forsyth canonical form v (n) + p3 v (n−3) + . . . + pn−1 v 0 + pn v = 0

(4.7)

with v ≡ v(u) and pk ≡ pk (u). It is obtained by the transformation x = F (u), y = F 0 (u)(n−1)/2 v where the inverse function u = F¯ (x) is Z 12q2 (x) 1 F¯ (x) = C1 exp z(x)dx + C2 with z 0 − z 2 − = 0. (4.8) 2 n(n2 − 1) The structure invariance group of (4.7) is u=

a1 u ¯ + a2 , a3 u ¯ + a4

v=

v¯ + L(u) (a3 u ¯ + a4 )n−1

where a1 , . . . , a4 are constants and L(u) is a linear combination of a fundamental system. Proof Due to Theorem 4.2, the desired transformation must be of the form x = F (u), y = F 0 (u)(n−1)/2 v. Substituting it into (4.6) (for the required derivatives see (B.16) of Appendix B) and imposing the constraint p2 (u) = 0 yields the equation  F 00 0 1  F 00 2 12q (F )F 02 2 − + =0 2 F0 F0 n(n2 − 1) for F (u) with F 0 = dF . For the inverse function u = F¯ (x) there follows du  F¯ 00 0 1  F¯ 00 2 12q2 (x) − − =0 (4.9) 0 0 ¯ ¯ 2 F F n(n2 − 1)

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165

¯ 00 ¯ where F¯ 0 = dF . This first order Riccati equation for z ≡ F¯ 0 has always a dx F solution. Any transformation of the structure invariance group of (4.7) is obtained by further specialization of (4.4). Substituting the corresponding relation G0 = n − 1 F 00 into the coefficient of y¯(n−2) (see (B.16) of Appendix B) leads n F0 G after some simplifications to  F 00 0 1  F 00 2 − = 0. F0 2 F0 From its general solution and after a further integration for obtaining G the final answer is C1 u + C2 C4 F = G= . , u + C3 (u + C3 )n−1 Introducing new constants by C1 =

a1 , a3

C2 =

a1 , a3

C3 =

a4 , a3

C4 =

1 an3

the above form of the structure invariance group corresponding to parameters a1 , . . . , a4 is obtained. The structure invariance group of the Laguerre-Forsyth canonical form has been given by Bouton [18], Chapter 5, equation (46). This is the first example in this chapter where the structure invariance group is a finite dimensional group. In order to determine the transformation to Laguerre-Forsyth canonical form explicitly, the Riccati equation (4.8) for z has to be solved. Substituting F¯ 0 = 1 2 the linear second order equation θ(x) θ00 +

6q2 θ=0 n(n2 − 1)

(4.10)

for θ is obtained. Then the transformation function F (u) is the inverse of Z dx u = F¯ (x) = . θ(x)2 The above equation (4.10) for θ shows that determining the Laguerre-Forsyth normal form is equivalent to solving a general second order linear equation as it has already been shown by Schlesinger [160], vol. II, page 198. For n = 2 this is equivalent to solving the original equation and no simplification of the solution procedure is obtained. This was to be expected because otherwise the general second order equation would be solvable which is known to be impossible. For equations of order higher than the second, however, the transformation to canonical form is a problem of more minor complexity than solving the given equation itself.

166 1 y = 0. Applying a general transformation Example 4.1 Consider y 000 + x of its structure invariance group yields y 000 +

(a1 a4 − a2 a3 )3 y = 0. (a1 x + a2 )(a3 x + a4 )5

In addition there are the invariance transformations x = x ¯, y = a5 y¯, and those corresponding to L(x). The third subject to be treated in this subsection are invariants of linear ode’s. A first result due to Laguerre is given next. Theorem 4.4 tion

(Laguerre 1879) Any third order linear homogeneous equay 000 + p1 y 00 + p2 y 0 + p3 y = 0,

pk ≡ pk (x)

(4.11)

allows the relative second order invariant Θ3 = p001 + 2p01 p1 − 3p02 + 94 p31 − 2p1 p2 + 6p3

(4.12)

w.r.t. to the structure invariance group x = F (u), y = G(u)v. It is transformed according to Θ3 (x)|x=F (u) · F 0 (u)3 = Θ3 (u). Proof The infinitesimal variations corresponding to the group F = u + f · δt, G = 1 are δx = f · δt, δy = 0, δy 0 = −f 0 y 0 · δt, δy 00 = −f 00 y 0 − 2f 0 y 00 and δy 000 = −f 000 y 0 − 3f 00 y 00 − 3f 0 y 000 . These values are substituted into the equation δy 000 + p1 δy 00 + p2 δy 0 + p3 δy + δp1 y 00 + δp2 y 0 + δp3 y = 0. The third derivative y 000 is substituted by those of lower order applying the given ode. The resulting expression can only vanish if the coefficient of each derivative including y itself vanishes. This leads to a system of linear equations from which the variations of the coeffcients may be determined. The result is δp1 = 3f 00 − p1 f 0 , δp2 = f 000 + p1 f 00 − 2p2 f 0 , δp3 = −3p3 f 0 . The variations of p0k and p00k are obtained from δpk by repeated application of Lie’s relation (Lie [114], page 670) δφ0 =

d d δφ − φ0 δx dx dx

(4.13)

which is valid for any φ. If invariants Φ of second order are to be determined, these variations have to be substituted into the equation  X  ∂Φ ∂Φ ∂Φ δpk + 0 δp0k + 00 δp00k = αΦ. ∂pk ∂pk ∂pk k=1,2,3

The coefficients of the various derivatives of f yield a linear first order system for Φ.

Equivalence and Invariants

167

For the group generated by G = (1 + g · δt)v, F = u, the infinitesimal variations are δx = 0, δy = gy ·δt, δy 0 = (g 0 y +gy 0 )·δt, δy 00 = g 00 y +2g 0 y 0 +gy 00 and δy 000 = g 000 y + 3g 00 y 0 + 3g 0 y 00 + gy 000 . By a similar analysis as above the variations δp1 = −3g 0 , δp2 = −4g 00 − 2p1 g 0 , δp3 = −g 000 − p1 g 00 − p2 g 0 are obtained. Together with its first and second derivatives they yield another linear first order system for Φ. The two systems have to be combined. After a lengthy calculation the following Janet basis is obtained. Φp003 = 0, Φp002 = 0, Θ3 Φp001 = Φ, Φp03 = 0, Θ3 Φp02 = −3Φ, Φp01 = 2p1 Φ,
Θ3 Φp3 = 6Φ, Θ3 Φp2 = 2p1 Φ, Θ3 Φp1 = 2P10 + 34 p21 − 2p2 Φ with Θ3 given by (4.12). This system has Φ = Θ3 as solution which is obtained by integration. 1 + 1
y 00 − 1 y = 0 which is equation 3.41 Example 4.2 Consider y 000 + x x2 x2
from the collection by Kamke with Θ3 (x) = − 16 6x4 − 94 x3 − 43 x2 + 38 x − 49 . x 1 , y = uv with v ≡ v(u) the equation becomes In new variables x = u  2 8  00  4 14  4 1 v 000 − 1 − v − − 2 v0 − − − v=0 u u u u2 u3 u4
with Θ3 (u) = − 14 49 u4 − 38 u3 + 43 u2 + 49 u − 6 . A simple calculation shows u that the two invariants obey the relation (4.12). Laguerre’s Theorem 4.4 does not require the equation to be in rational normal form. As explained in the introduction to this chapter however, the complexity of any calculation involving differential equations increases significantly if the number of nonvanishing coefficients increases. This is particularly true for any kind of invariants. Therefore from now on in this section any linear ode will be assumed in rational normal form. Theorem 4.5 (Schlesinger 1897) Any linear homogeneous equation in rational normal form y (n) + q2 y (n−2) + . . . + qn−1 y 0 + qn y = 0 allows a series of n − 2 relative invariants Θk , k = 3, . . . , n w.r.t. to the group x = F (u), y = F 0 (u)(n−1)/2 v transforming like Θk (u) = F 0 (u)k Θk (x)|x=F . Explicitly the three lowest invariants are Θ3 = q3 −

n−2 0 q , 2 2

168 Θ4 = q4 −

n−3 0 (n − 2)(n − 3) 00 1 5n + 7 (n − 2)(n − 3) 2 q2 , (4.14) q + q2 − 2 3 10 10 n + 1 n(n − 1)

3(n − 3)(n − 4) 00 (n − 2)(n − 3)(n − 4) 00 4 0 q3 − q2 Θ5 = q5 − n − 2 q4 + 28 84 + 13 − 71 7n n+1

(n − 3)(n − 4) 1 7n + 13 (n − 2)(n − 3)(n − 4) q q 0 . q2 q3 + 14 2 2 n+1 n(n − 1) n(n − 1)

The proof may be found in Schlesinger [160], vol. II, page 147. If p1 = 0, p2 = q2 and p3 = q3 are substituted into (4.12), it is identical to Θ3 defined in (4.12), up to a constant factor. This example shows the considerable simplification due to rational normal form. For higher order invariants the difference in size is even more prominent. Theorem 4.6 normal form

(Forsyth 1888) Any linear homogeneous equation in rational y (n) + q2 y (n−2) + . . . + qn−1 y 0 + qn y = 0

allows a series of n − 2 relative invariants Φk , k = 3, . . . , n w.r.t. to the group x = F (u), y = F 0 (u)(n−1)/2 v transforming like Φk (u) = F 0 (u)2 Φk (x)|x=F . In terms of the invariants Θk they are Φk (x) = 2k

 Θ0 2 Θ00k 24k 2 q2 . − (2k + 1) k − Θk Θk n(n2 − 1)

Combining the Θk and the Φk , the following series of absolute invariants Θj,k ≡

Θjk Θkj

and

Ψk ≡

Φkk Θ2k

is obtained. The invariants Φk are obtained by differentiation and substitution from the invariants Θk . The details may be found in the original article by Forsyth [47], a good account is also given in the book by Schlesinger [160], part II, Chapter 6, page 200. Equivalence of Second Order Equations. In order to proceed further with identifying equivalence classes of linear ode’s, equations of fixed order are considered. The complete answer for second order linear ode’s is given next. Theorem 4.7 Any linear second order ode y 00 + p1 y + p2 y = 0 is equivalent to v 00 = 0. As a consequence all second order linear ode’s form a single equivalence class with canonical representative v 00 = 0. Proof At first the given equation is transformed into its rational normal form y 00 + py = 0 with p = p2 − 12 p01 − 14 p21 . It remains to be shown that

Equivalence and Invariants

169

v 00 = 0 may be transformed into it by a transformation u = f (x), v = g(x)y. According to (B.15) the second derivative is  g 00  g0 f 00  f 00 g 0  i g h y . v 00 = 02 y 00 + 2 − 0 y 0 + − 0 g g f f g f It yields the equation y 00 + py = 0 if 2

g0 f 00 g 00 f 00 g 0 − 0 = 0, − 0 = p. g g f f g

g0 f 00 g 00 g 0
0 g 0
2 Substituting g = 2 0 and g = g + g into the latter equation, the f g0 Riccati equation z 0 − z 2 − p2 + 12 p01 + 14 p21 = 0 for z ≡ g is obtained. Solving for f and g the expressions Z  Z  Z  f = C1 exp 2 zdx dx + C2 , g = C3 exp zdx follow; z is a solution of the above Riccati equation. Example 4.3 Let {1, u} and {y1 , y2 } be fundamental systems for v 00 = 0 y and y 00 + p1 y 0 + p2 y = 0 respectively. The transformation u = σ ≡ y12 and y v = ρ ≡ y2 transforms the latter into the former. Correspondingly v 00 = 0 is transformed into the latter equation in the form (2.3). This result has been known for a long time; see for example Schlesinger [160], vol. II, page 184, equation (8). It is mainly of theoretical interest because it does not help in solving any second order equation. Solving the Riccati equation that occurs at the end of the proof of Theorem 4.7 comes down to solving the original linear second order ode. Numerous higher transcendents or special functions are defined in terms of linear second order ode’s, e. g. Bessel functions, hypergeometric functions, etc. In applications there often arises the problem of recognizing whether the solution of a given differential equation may be expressed in terms of particular special functions. The answer may be obtained by the following result which generalizes the above theorem. Theorem 4.8 A linear second order ode y 00 + p(x)y = 0 is transformed into v 00 + q(u)v = 0 by x = f (u), y = f (u)01/2 v if a function f (u) may be found that obeys  f 00 0 1  f 00 2 − +2f 02 p(f ) = 2q(u). (4.15) 2 f0 f0 This third order ode has always a six-parameter group of symmetries. Proof By Theorem 4.2 the desired equivalence transformation must have the form (4.4) for n = 2. Substituting it into y 00 + p(x)y = 0, applying (B.15)

170 and equating the result to v 00 + q(u)v = 0 leads immediately to the above equation for f . A simple calculation shows that the conditions for case ii) of Theorem 5.15 are satisfied, i.e., (4.15) belongs to symmetry class S63 . In order to determine an equivalence transformation to any equation defining a special function by Theorem 4.8, a rational normal form must be known for it. The most important ones are given next. The defining equation is written in terms of the independent variable z and dependent variable w. For the rational normal forms the dependent variable is w. ¯ Further details may be found in Abramowitz [2]. It should be emphasized again that the given rational normal forms are not unique. The Bessel equation  ν2  1 w00 + w0 + 1 − 2 w = 0 z z

(4.16)

with fundamental system J±ν (z). A rational normal form is  √ ν 2 − 14  w ¯ 00 + 1 − ¯ = zw. w ¯ = 0 with w z2

(4.17)

The hypergeometrical equation w00 +

(α + β + 1)z − γ 0 αβ w + w = 0. z(z − 1) z(z − 1)

(4.18)

A rational normal form is w ¯ 00 −

[(α − β)2 − 1]z 2 + 2[(α + β − 1)γ − 2αβ]z − γ(γ − 2) w ¯=0 4z 2 (z − 1)2

(4.19)

with w = z γ/2 (z − 1)(α+β+1−γ)/2 w. ¯ The Legendre equation w00 +

ν(ν + 1) 2z w0 − 2 w = 0. z2 − 1 z −1

(4.20)

A rational normal form is w ¯ 00 −

p ν(ν − 1)(z 2 − 1) − 1 w ¯ = 0 with w ¯ = z 2 − 1w. (z 2 − 1)2

(4.21)

Weber’s equation w00 − zw0 − aw = 0.

(4.22)

A rational normal form is
w ¯ 00 − 14 [z 2 + 2(a − 1)]w ¯ = 0 with w ¯ = exp − 14 z 2 w.

(4.23)

The confluent hypergeometric equation or Whittaker’s equation is already in rational normal form. 1 k 4m2 − 1  y 00 − − + y = 0. (4.24) 4 x 4x2

Equivalence and Invariants

171

Despite the large symmetry group of equation (4.15), the solution algorithm for this symmetry type described in Chapter 7 on page 344 is of no help here. The reason is that it requires solving the second order linear ode’s for which the equivalence transformation is desired. Details are discussed in Exercise 4.8. In order to proceed, the problem has to be further specialized. It is based on the observation that equivalence to an equation defining a special function is only of interest if the transformation is simple in some sense, e. g. it should be rational or algebraic. For equivalence to Bessel’s equation the following result is useful. Lemma 4.1 equations

The transformations z = axα , w ¯ = (αaxα−1 )1/2 y¯ generate the

i
− α2 ν 2 12 y¯ = 0 x from the rational normal form (4.17) of Bessel’s equation. y¯00 +

h

α2 a2 x2α−2 +

1 4

(4.25)

It is easily proved by applying the formulas (B.15) given in Appendix B to the suggested transformations. Let y 00 + q1 y 0 + q2 y = 0 be a given Requation,
y¯00 + q(x)¯ y = 0 its rational normal form obtained by y = exp − 21 q1 dx y¯. If q(x) has the form of the square bracket in (4.25), system of R √ a fundamental the original equation in y(x) may be written as x exp (− 12 q1 dx)w1,2 (axα ) if w1,2 (z) is a fundamental system of (4.16). It turns out that the class of transformations admitted in Lemma 4.1 covers virtually all equivalence problems for Bessel’s equation listed in Kamke’s collection. A few examples are given next. 1 y0 + 1 y = 0 Equation 2.347 from Kamke’s collection y 00 + x x4
1 1 1 00 has the rational normal form y¯ + 4 + 2 y¯ = 0 if y = √ y¯ is chosen. x x 4x The above lemma yields α = −1, a = ±1 and ν = 0. With a = −1 there y¯ 1, w follows z = x ¯ = x . Finally the full transformation from the originally given equation to (4.16) with ν = 0 is x = z1 , y = w, consequently, it has a 1
and Y 1
. fundamental system J0 x 0 x Example 4.4

Example 4.5 Let the irreducible Airy equation y 00 + xy = 0 be given. It is already in rational normal form. By Lemma 4.1 α = 23 , a = 23 and ν = 31 is Consequently, a fundamental system for the original equation is
√ obtained. xJ±1/3 23 x3/2 . For Weber’s equation the equivalent of Lemma 4.1 is given next. Again this class of transformations covers all cases listed in Kamke’s collection. Lemma 4.2 The transformation z = axα , w ¯ = (αaxα−1 )1/2 y¯ generates the equation h i 2 (4.26) y¯00 + 41 α2 a4 x4α−2 + 12 α2 a2 (k − 1)x2α−2 + α −2 1 y¯ = 0 4x from the rational normal form (4.17) of Bessel’s equation.

172 Equivalence of Third Order Equations. For linear equations of third order it occurs for the first time that there is more than a single equivalence class. This is because a linear homogeneous equation of order n contains n coefficient functions. Only two of them may be transformed into each other by the two functions F (u) and G(u) contained in the structure invariance group given in Theorem 4.1. Obviously this is only possible for n = 2. The next theorem applies the invariants defined in the preceding subsection in order to identify all equivalence classes of linear third order equations. Theorem 4.9

Let two third order linear homogeneous ode’s

y 000 + A(x)y 0 + B(x)y = 0 and v 000 + P (u)v 0 + Q(u)v = 0

(4.27)

in rational normal form be given. The four alternatives below define equivalence classes, canonical forms and give the equations that are obeyed by the transformation function x = F (u), y = F 0 (u)v to canonical form, or its inverse u = F¯ (x), v = F¯ 0 (x)y. The required invariants  0 2 Θ00 Φ3 Θ −9A and Ψ3 = 32 Θ3 = B − 12 A0 , Φ3 = 6 Θ3 − 7 Θ3 3 3 Θ3 are obtained from the preceding Theorems 4.14 and 4.6 for n = 3. E13 : Θ3 = 0, canonical form v 000 = 0, F¯ (x) obeys  F¯ 00 0  ¯ 00 2 F 3 − − 12 A = 0. 2 F¯ 0 F¯ 0

(4.28)

1 E23 : Θ3 6= 0, Φ3 = 0, canonical form v 000 + v = 0, F¯ obeys F¯ 0 = Θ3 (x) 3 .

E33 (c) : Θ3 6= 0, Φ3 6= 0, Ψ3 = const, canonical form v 000 +cv 0 +v = 0, c 6= 0. The constant c parametrizes equivalence classes, while F (u) is determined by integration from F 02 · Φ3 (x)|x=F = −9c. E43 (Q) : If none of the preceding three cases applies, the two equations (4.27) are equivalent to each other if both Θ3 (x)|x=F F 0 (u)3 = Θ3 (u) and Ψ3 (x)|x=F = Ψ3 (u) are valid. The equation for y is equivalent to v 000 + Q(u)v = 0 if F¯ (x) is determined from (4.28), and its inverse F (u) is substituted into the second equation of (4.29) from which Q is obtained. Proof Applying a general transformation of the structure invariance group x = F (u) and y = F 0 (u)v to the above equation for y(x) yields h  00 2 i 000 v 000 + 2 F 0 − 3 F 0 +A(F )F 02 v 0 F F h (4)  00 3 i 000 00 F + A(F )F 0 F 00 + B(F )F 03 v = 0. + F 0 +3 F0 −4 F 02 F F F

Equivalence and Invariants

173

In order to generate the equation for v(u), F (u) has to obey the constraints

F (4) F0

 2 F 000 − 3 F 00 + 12 A(F )F 02 = 12 P, 0 0 2 F F  00 3 000 00 F + A(F )F 0 F 00 + B(F )F 03 = Q. +3 F0 −4 F 02 F F

(4.29)

Reduction of the second equation w.r.t. the first one yields i h F 0 (u)3 = Θ3 (u) B(F ) − 12 A0 (F ) F 03 = Q − 12 P 0 or Θ3 (x) x=F

(4.30) dA(F ) . If Θ3 (x) = Θ3 (u) = 0, the second equation of (4.29) where A (F ) = dF is the derivative of the first and the system reduces to a single third order ode for F which has always a solution. If P = Q = 0 is chosen for the canonical form equation, F (u) satisfies 0



F 00 F0

0

− 21



F 00 F0

2

+ 21 A(F )F 02 = 0

from which the Riccati equation (4.28) for the inverse function F¯ (x) is obtained. If Θ3 (x) = 0 and Θ3 (u) 6= 0 or Θ3 (x) 6= 0 and Θ3 (u) = 0, the above system (4.29) for F is inconsistent and the two equations for y and v cannot be equivalent to each other. Consequently, the condition Θ3 = 0 characterizes the single equivalence class E13 . If both Θ3 (x) 6= 0 and Θ3 (u) 6= 0, reducing the first equation of (4.29) w.r.t. (4.30) yields 6

 Θ0 (u) 2  Θ0 (F ) 2 h Θ00 (F ) i Θ003 (u) 3 3 −7 −9P = 6 3 −7 −9A(F ) F 02 Θ(u) Θ3 (u) Θ(F ) Θ3 (F )

where Θ03 (F ) =

dΘ3 (F ) , or dF Φ3 (u) = Φ3 (x)|x=F · F 0 (u)2 .

(4.31)

If Φ3 (x) = Φ3 (u) = 0, F is determined by the first order ode (4.30) which has always a solution. If P = 0, Q = 1 is chosen for the canonical form equation, 1 θ3 (u) = 1 and the equation F¯ (x)0 = Θ3 (x) 3 is obtained. If Φ3 (x) = 0 and Φ3 (u) 6= 0 or Φ3 (x) 6= 0 and Φ3 (u) = 0, the system for F is again inconsistent and the two equations are not equivalent. Consequently, Θ3 = 0, Φ3 = 0 characterize another equivalence class E23 . If both Φ3 (x) 6= 0 and Φ3 (u) 6= 0, consistency of (4.30) and (4.31) requires Ψ3 (u) = Ψ3 (x)|x=F . If both Ψ3 (x) and Ψ3 (u) are constant, a possible choice for the canonical form is Q = 1 and P = c = const, the value of c is determined by Ψ3 (u) = −729c3 = Ψ3 (x). In general, two canonical equations with c 6= c¯ are not equivalent, i.e., any given value of c determines an equivalence class

174 E33 (c). Substituting the corresponding value for Φ3 (u) = −9c into (4.31) leads to the equation given for F given above. If both invariants Ψ3 (x) and Ψ3 (u) are not constant, the transformation function F (u) has to be determined as a solution of the system (4.29) or equivalently (4.30) and (4.31). Its consistency condition Ψ3 (u) = Ψ3 (x)|x=F is necessary for equivalence. If it is satisfied, the validity of (4.30) is sufficient for equivalence because in this case (4.31) is a consequence of it. Choosing P = 0 for the canonical form equation, the first equation determines F (u); as a consequence F¯ (x) satisfies the same equation as in case i). Q is obtained by substituting the solution for F (u) into the second equation of (4.29). The canonical form for E43 (Q) is actually the Laguerre-Forsyth canonical form of Theorem 4.3, the Riccati equation of the above theorem corresponds to the equation (4.8) for z given there. In order to decide equivalence of two ¯ it is not necessary to solve a Riccati given equations in E33 (Q) and E33 (Q), equation. Rather a possible transformation function is determined from the condition Ψ3 (u) = Ψ3 (x)|x=F for the invariants which is a purely algebraic problem. If the resulting function F (u) satisfies (4.30) and (4.31), equivalence is established, otherwise the two equations are not equivalent to each other. The following examples illustrate this case. Example 4.6 Let the two equations y 000 + xy 0 + y = 0 with Θ3 (x) = 12 , Φ3 (x) = −9x, Ψ3 (x) = −2916x3 and v 000 + 15 v 0 − 36 v = 0 with Θ3 (u) = − 1 6 , u u 2u 3 Φ3 (u) = − 95 and Ψ3 (u) = − 2916 3 be given, i.e., they belong to E4 (Q). By u u C Theorem 4.9 the differential equation F 0 = − F u with the solution F = u is 1 , i.e., the two obtained. The condition for the invariants Ψ3 leads to F = u 1 and y = − v establish equations are in the same equivalence class; x = u u2 the transformation. Let the equations y 000 + xy 0 + y = 0 like in the preceding 2 example and v 000 + v 0 + uv = 0 with Θ3 (u) = u, Φ3 (u) = − 9u 2+ 7 and u (9u2 + 7)3 Ψ3 (u) = − be given. By Theorem 4.9 the constraints u8 Example 4.7

2

(9u2 + 7)3 18u3 F Ce9u 0 9 F = , F (u) = −→ F = 2916u8 9u2 + 7 (9u2 + 7)7 3

are obtained which are inconsistent for any value of the integration constant C. As a consequence the two equations are not equivalent to each other. Example 4.8 The equation y 000 − y 00 = 0 with rational normal form 1 0 2 2 y − 3 y − 27 y = 0 has invariants Θ3 = − 27 , Φ3 = 3 and Ψ3 = 19683 4 . Transfor√ √ 3 3 mation to canonical form v 000 − 2 2v 0 + v = 0 is achieved by F (u) = − 23 3 4u, √ 1 y 00 = 0 with rational normal i.e., it belongs to E33 (− 32 3 2). Similarly, y 000 − x 000

Equivalence and Invariants

175

form y 000 − 4 2 y 0 + 16 3 y = 0 and invariants Θ3 = − 20 3 , Φ3 = 212 and Ψ3 = 3x 27x 27x x 21 ), because it is transformed 137781400 belongs to equivalence class E33 (− 2 √ 3 50 √ 21 3 3 000 0 √ to canonical form v − 2 3 50 v + v = 0 by F (u) = exp (− 10 50u). In the next chapter it will be shown how the various equivalence classes described in Theorem 4.9 combine into symmetry classes. In this context equations with constant coefficients play an important role. They are the subject of the next example. Example 4.9 For linear ode’s with constant coefficients the results of Theorem 4.9 may be described more explicitly. The generic equation of 3 000 0 the equivalence a. By √ class E1v is y + ay = 0 with a nonzero constant 000 u x = √ log i a, y = u it is transformed into the canonical form v = 0. i a The generic equation for the equivalence class E23 is y 000 + by = 0 where b 6= 0 q
1/3 q
1/3 u, y = v it is transformed into is again a constant. By x = b b v 000 + qv = 0 where q is an arbitrary constant. The canonical form v 000 + v = 0 1
1/3 u, y = 1
1/3 v. The generic equation for case E 3 (c) is obtained by x = a 3 a has the form y 000 + ay 0 + by = 0 with both a, b nonvanishing constants. It cannot be transformed into an arbitrary equation of the same kind, only one of its coefficients may be chosen at will, e. g. the coefficient of v may be made a v 0 + v = 0. to unity as in the preceding case with the result v 000 + 2/3 b In later chapters different canonical forms will be applied for equivalence classes E23 and E33 (c). They are described in the subsequent lemma. Lemma equation y 000 + cy 0 + y = 0 with c 6= 0 and √ 4.3 A third order 3 3 000 c 6= − 2 2 is equivalent to v − (a + 1)v 00 + av 0 = 0, if the constants a and c are related to each other by (a − 21 )2 (a + 1)2 (a − 2)2 c3 +

27 2 4 (a

− a + 1)3 = 0.

(4.32)

The transformation between the two equations is achieved by x = αu and y = eβu v where α=

1 9

(a + 1)(a − 2)(2a − 1)c , a2 − a + 1

β = − 13 (a + 1).

√ If c = 0 it is equivalent to v 000 −(a+1)v 00 +av 0 = 0 with a = 21 (1±i 3) whereby √ α = 31 (a + 1), β = − 13 (a + 1). If c = − 23 3 2 it is equivalent to v 000 − v 00 = 0 √ whereby α = − 13 3 2, β = − 13 . Proof By Corollary 4.1 the transformation between the two linear ode’s must have the form x = αu, y = eβu v. Substitution into the first equation yields the algebraic system 3β + a + 1 = 0, cα2 + 3β 2 − a = 0, α3 + cα2 β + β 3 = 0

176 from which α and β are obtained by elimination if the constraint (4.32) is satisfied. The calculations in the remaining cases are similar. The relation between equivalence classes and the value of c in the canonical form representative y 000 + cy 0 + y = 0 is not one-to-one. The same is true for the canonical form y 000 − (a + 1)y 00 + ay 0 = 0 in the above lemma. These relations are discussed in more detail in the Exercises 4.6 and 4.7.

4.2

Nonlinear First Order Equations

The first result in this section is mainly of theoretical interest. It shows that point symmetries will be of limited help in solving first order ode’s because they are members of a single equivalence class. Consequently, they all share the same symmetry type. Theorem 4.10 Any quasilinear first order equation y 0 + r(x, y) = 0 is equivalent to v 0 = 0, i.e., all first order equations of this kind form a single equivalence class under point transformations for which v 0 = 0 may be chosen as a canonical form. The point transformation x = φ(u, v), y = ψ(u, v) transforms the ψ + r(φ, ψ)φu given equation into v 0 + u = 0. The desired canonical form is ψv + r(φ, ψ)φv obtained if ψu + r(φ, ψ)φu = 0. This equation always has a solution. Proof

Although this result guarantees that an equivalence transformation to canonical form v 0 = 0 does exist for any given first order ode, it does not provide a method for finding such a transformation. In general it comes down to solving the given equation as is obvious from rewriting the condition in the dψ proof as + r(φ, ψ) = 0. In Exercise 4.2 the reverse transformation from dφ the canonical form to a general first order equation is considered. 4 y + 2 = 0 has the general The Riccati equation y 0 + y 2 + x x2 x(xy + 1) 1 2 solution y = x + C − x . It yields the first integral xy + 2 = C. According x(xy + 1) to Exercise 4.2 this yields the canonical variables v ≡ ρ(x, y) = xy + 2 and 1 one obtains u ≡ σ(x, y) where σ(x, y) may be chosen arbitrarily. For u = x 1 , y = u(1 − 2uv) . For u = x the transformation is the transformation x = u uv − 1 x = u, y = 2v − u . u(u − v) Example 4.10

From this result follows that in order to proceed, more special classes of

Equivalence and Invariants

177

equations have to be considered, e. g. by limiting the admitted transformations to the corresponding structure invariance groups. The two major classes are Riccati’s equation containing a quadratic term in the dependent variable, and Abel’s equation with a cubic term. Riccati’s Equation. Algorithms for finding special solutions of these equations have been described already in Chapter 2. Here they will be reconsidered in the context which is the subject of this chapter. Its structure invariance is determined first. Theorem 4.11

The structure invariance group of Riccati’s equation

y 0 + a2 y 2 + a1 y + a0 = 0,

ak ≡ ak (x) f or k = 0, 1, 2

(4.33)

is x = F (u), y = G(u)v + H(u) where v ≡ v(u) and F , G and H are undetermined functions of its argument. Proof A general point transformation from x and y to new variables u and v with v ≡ v(u) x = φ(u, v), y = ψ(u, v), y 0 =

ψ u + ψv v 0 φu + φv v 0

generates the equation v0 +

(b2 ψ 2 + b1 ψ + b0 )φu + ψu =0 (b2 ψ 2 + b1 ψ + b0 )φv + ψv

(4.34)

where bk ≡ ak (φ). In order to avoid the occurrence of an unspecified dependence on v via the coefficients ak , φv = 0 is required, i.e., φ = F (u). The denominator must be independent of v, this requires ψv to be a function of u alone, i.e., it must be linear in v and has the form ψ = G(u)v + H(u). Under these constraints (4.34) simplifies to h G0 i F0 H0 + (2b2 H + b1 )F 0 v + (b2 H 2 + b1 H + b0 ) + = 0, G G G (4.35) i.e., it has the desired structure. v 0 + b2 F 0 Gv 2 +

Theorem 4.12 The rational normal form of Riccati’s equation (4.33) is y 0 + y 2 + A(x) = 0. It has the structure invariance group x = F (u), y =

1  1 F 00  v+ . 0 2 F0 F

(4.36)

Proof By Theorem 4.11 the desired transformation must have the general form x = F (u), y = G(u)v + H(u). From (4.35) the constraints b2 F 0 G = 1,

G0 + (2b2 H + b1 )F 0 = 0 G

178 are obtained with bk ≡ ak (F ). Eliminating G and H yields for the transformation to rational normal form x = F, y =

b2 F 00 + b02 F 0 − b1 b2 F 02 v . 0 + b2 F 2b22 F 02

(4.37)

If F ≡ F (u) is in the base field of (4.33), the same is true for y and y 0 obtained from it. On the other hand, any additional constraint on the term which is free of the dependent variable in order to obtain a particular form A(x) requires solving a Riccati equation for H which in general is not possible within the base field. This proves the first part. Any transformation of the structure invariance group of y 0 + y 2 + A = 0 must have the general form (4.35) with the additional constraints F 0 G = 1, G0 + 2HF 0 = 0. This yields (4.36). G Sometimes it occurs that the rational normal form allows one to obtain the solution immediately as the following examples show. Example 4.11

The two equations from Kamke’s collection

4 y + 2 = 0 and no.1.140 : y 0 + y 2 + x x2 1 0 2 no.1.165 : y + 2 y − 4x2 + 1 y + 2x 4− 1 = 0 2x − x 2x − x 1 . have rational normal form v 0 + v 2 = 0 with general solution v = u + C If the rational normal form does not contain the independent variable explicitly, the solution may be obtained as explained on page 18. Two examples are given next. Example 4.12 Equation 1.15 y 0 +y 2 −2x2 y +x4 −2x−1 = 0 from Kamke’s collection has rational normal form v 0 + v 2 − 1 = 0 with a special solution 4 y + 2 = 0 with rational normal form v = 1. Equation 1.140 y 0 + y 2 + x x2 v 0 + v 2 = 0 has the special solution v = 0. In either case, the general solution may be obtained by Corollary 2.1 or by direct integration. Corollary 4.2 Any Riccati equations y 0 + y 2 + A(x) = 0 and v 0 + v 2 + P (u) = 0 are equivalent w.r.t. to the structure invariance group (4.36). Proof

Applying a transformation (4.36) to the first equation yields

1  F 00 0 1  F 00 2 − +A(F )F 02 = 0. 2 F0 4 F0  00 0  00 2 This represents the second equation if 12 F 0 − 14 F 0 +A(F )F 02 = F F P (u). Because this third order ode always has a solution, equivalence is always assured. v0 + v2 +

Equivalence and Invariants

179

This result is an immediate consequence of Theorem 4.10. It shows that also for Riccati equations an equivalence transformation in general cannot be found. Abel’s Equation. The simplest first order equation that cannot be transformed easily into a linear one as is true for Riccati’s equation is the equation introduced about 200 years ago by the famous Norwegian mathematician Abel. It has the form y 0 + a3 y 3 + a2 y 2 + a1 y + a0 = 0

(4.38)

with ak ≡ ak (x) for k = 0, . . . , 3, a3 6= 0. The more general equation y0 +

f3 y 3 + f2 y 2 + f1 y + f0 =0 y+g

(4.39)

which is usually called Abel’s equation of second kind may be reduced to (4.38) by changing the dependent variable y = 1 − g with the result v(x) a3 = f3 g 3 − f2 g 2 + f1 g − f0 , a2 = g 0 − 3f3 g 2 + 2f2 g − f1 , a1 = 3f3 g − f2 , a0 = −f3 .

(4.40)

If in (4.39) the numerator is of degree two in y, i.e., if f3 = 0, it follows that a0 = 0. As usual by now, the structure invariance group of (4.38) will be identified first. Although the result has been known for a long time, for example to Appell and Painlev´e, it has never really been proved. Therefore it is derived next. Theorem 4.13 is

The structure invariance group of Abel’s equation (4.38) x = F (u),

y = G(u)v + H(u)

(4.41)

where v ≡ v(u) and F , G and H are undetermined functions of its argument. Proof The first part of the proof is identical to the proof of Theorem 4.11 with the result φ = F (u) and ψ = G(u)v+H(u). Applying this transformation to (4.38) yields  0  v 0 + b3 F 0 G2 v 3 + (3b3 H + b2 )F 0 Gv 2 + 3b3 H 2 + 2b2 H + b1 + G0 F 0v F G 0 H0 = 0 +(b3 H 3 + b2 H 2 + b1 H + b0 ) F + G G (4.42) with bk (u) ≡ ak (F ), i.e., it has the desired structure. Abel’s equation (4.38) contains four coefficient functions, whereas there are only three in the structure invariance group (4.41). This suggests that Abel’s

180 equation may be transformed into an equation with only a single nonconstant coefficient involved. This transformation was performed by Appell [6] with the result y 0 + y 3 + J(x) = 0. However, in general J(x) is not contained in the base field; it is left as an exercise to determine it explicitly. The optimal result that may be achieved without leaving it is obtained next. Theorem 4.14 There are two different rational normal forms of Abel’s equation (4.38). Let A ≡ A(x) and B ≡ B(x) be rational functions of a0 , . . . , a3 and its derivatives, A 6= 0. i) y 0 +Ay 3 +By = 0 with structure invariance group x = F (u), y = G(u)v. ii) y 0 + Ay 3 + By + 1 = 0 with structure invariance group x = F (u), y = F 0 (u)v. Proof If any transformation (4.41) is applied to (4.38), it assumes the form (4.42). The choice H = − b2 entails the vanishing of the coefficient of 3b3 v 2 . As a consequence applying the transformation a2 1  a2 0 y=v− , y0 = v0 − 3a3 3 a3 which leaves the independent variable unchanged, Abel’s equation assumes the form v 0 + b3 v 3 + b1 v + b0 = 0 with b3 = a3 , b1 = a1 −

a22 a1 a2 1  a2 0 2a32 − . , b0 = a0 − + 3a3 3a3 3 a3 27a23

(4.43)

If b0 = 0, the first alternative is obtained. Otherwise introducing again a new function w by v = b0 w, v 0 = b0 w0 + b00 w leads to w0 + Aw3 + Bw + 1 = 0 with A = b20 b3 , B = b1 +

b00 . b0

This is the canonical form for case ii). For the structure invariance group of either case the terms independent of v in (4.42) lead immediately to the constraint H = 0. If b0 = 1 the additional constraint G = F 0 is necessary in order not to change this value. The rational normal form of case i) is actually a Bernoulli equation. It is a pleasant feature that this special case is identified without any additional effort. As is well known, if the new function v is replaced by 12 the linear v equation v 0 − 2Bv − 2A = 0 is obtained that is easily solved in terms of quadratures with the result Z  Z h Z  i v = exp 2 Bdx) 2 A exp (−2 Bdx dx + C . (4.44)

Equivalence and Invariants

181

As will be seen later on, this is a consequence of a two-parameter symmetry group that may always be determined explicitly for a Bernoulli equation. 1 = 0 from Equation 1.146 y 0 − x2 y 3 + 3y 2 − 12 y + 2x − x x4 Kamke’s collection has rational normal form v 0 − u2 v 3 + 22 v = 0, i.e., it is u a Bernoulli equation the solution of which may be obtained by integration as explained above. Example 4.13

The invariants of Abel’s equation with respect to the group (4.41) were given for the first time by Liouville [119]. He did not mention, however, how he obtained them. For the applications in this book only the two lowest invariants derived in the next theorem are needed. Theorem 4.15

(Liouville 1887) For Abel’s equation y 0 + a3 y 3 + a2 y 2 + a1 y + a0 = 0

(4.45)

where ak ≡ ak (x) a relative invariant w.r.t. its structure invariance group x = F (u), y = G(u)v + H(u) with v ≡ v(u) is Φ3 = a2 a03 − a02 a3 + 3a0 a23 − a1 a2 a3 + 92 a32 .

(4.46)

By differentiation the series of higher order relative invariants Φ2m+1 = a3 Φ02m−1 − (2m − 1)[a03 + 31 a22 − a1 a3 ]Φ2m−1

(4.47)

for m = 2, 3, . . . is obtained. To the invariant Φk the weight k is assigned. Denoting the coefficients of the original and the transformed equation collectively by a and b, the transformation law between the invariants is Φk (u) = F (u)0k · G(u)k · Φk (x)|x=F (u) . Rational expressions of these relative invariants with the property that the sum of the products of their order and the corresponding exponent are the same in the numerator and the denominator are absolute invariants. Proof The expression (4.46) for the first order invariant Φ3 will be derived in detail. The infinitesimal variations corresponding to the group F = u+f ·δt, G = 1, H = 0 are δx = f · δt, δy = 0, δy 0 = −f 0 y 0 · δt. Substituting these values into the variation of Abel’s equation yields δy 0 + (3a3 y 2 + 2a2 y + a1 )δy + δa3 y 3 + δa2 y 2 + δa1 y + δa0 = −f 0 y 0 + δa3 y 3 + δa2 y 2 + δa1 y + δa0 = 0. If in the latter expression y 0 is eliminated by means of Abel’s equation and the result is separated w.r.t. to y, a system of equations is obtained from which the variations of the coefficients may be determined as δak = −f 0 ak · δt. The variations of the derivatives a0k follow from these expressions by means of the

182 identity (4.13) with the result δa0k = −(ak f 00 + 2ak f 0 ) · δt. Accordingly a relative invariant Φ for this group has to obey the system of equations a3 Φa03 + a2 Φa02 + a1 Φa01 + a0 Φa00 = 0, 2a03 Φa03 + 2a02 Φa02 + 2a01 Φa01 + 2a00 Φa00 + a3 Φa3 + a2 Φa2 + a1 Φa1 + a0 Φa0 = 3Φ. (4.48) The term 3Φ at the right hand side originates from the variation of the factor F 0m that yields the contribution mf 0 · δt. It turns out that only m = 3 allows a nontrivial solution for Φ. For the group generated by G = (1 + g · δt)v, F = u and H = 0, the infinitesimal variations are δx = 0, δy = gy · δt, δy 0 = (g 0 y + gy 0 ) · δt. By a similar analysis to the above the variations δa0 = ga0 , δa1 = −g 0 , δa2 = −ga2 , δa3 = −2ga3 , δa00 = g 0 a0 + ga00 , δa01 = −g 00 , δa02 = −ga02 − g 0 a2 , δa3 = −2g 0 a3 − 2ga03 of the coefficients and its derivatives are obtained. They yield the system Φa01 = 0,

2a3 Φa03 + a2 Φa02 − a0 Φa00 + Φa1 = 0,

2a03 Φa03 + a02 Φa02 − a00 Φa00 + 2a3 Φa3 + a2 Φa2 − a0 Φa0 = 3Φ

(4.49)

for the invariants. Finally the group F = u, G = 1 and H = h · δt generates the variations δx = 0, δy = h · δt and δy 0 = h0 · δt of the variable and the first derivative. The variations of the coefficients and its derivatives are δa0 = ha0 , δa1 = −h0 , δa2 = −a2 h, δa3 = −2ha3 , δa00 = h0 a0 + ha00 , δa01 = −h00 , δa02 = −a02 h − a2 h0 , δa3 = −2h0 a3 − 2ha03 . They yield the system Φa00 = 0,

3a3 Φa02 + a2 Φa01 + Φa0 = 0,

3a03 Φa02 + 2a02 Φa01 + 3a3 Φa2 + 2a2 Φa1 + a1 Φa0 = 0.

(4.50)

The full system comprising (4.48), (4.49) and (4.50) determines the desired invariants. It is arranged in grlex term order with ak > aj for k > j. Due to the group properties of the transformations of x and y it is a complete system. As a consequence, a Janet basis is obtained by purely algebraic autoreduction steps with the result 3a2 2 a3 Φ = 0, Φa0 − Φ 3 Φ = 0, Φa1 + aΦ 3 3 3a1 a3 − 2a22 − 3a03 6a0 a3 − a1 a2 − a02 Φ = 0, Φ − Φ = 0, a 3 3Φ3 Φ3 a3 Φ = 0, Φ 0 − a2 Φ = 0 Φa00 = 0, Φa01 = 0, Φa02 + Φ a3 Φ3 3

Φa2 −

Equivalence and Invariants

183

with Φ3 given by (4.46). It is easily seen that Φ ≡ Φ3 solves this system. The higher order invariants (4.47) may be verified by differentiation and elimination. If in a rational expression involving the invariants the powers of F (u)0 in the numerators and the denominator are the same, F (u)0 drops from this expression. The same is true for G(u), i.e., a rational expression with this property is an absolute invariant. Due to the fact that the structure invariance group of Abel’s equations in rational normal form is a subgroup of (4.41), their invariants may be obtained by specialization of the invariants determined in the above theorem. Of particular interest for solving equivalence problems are the lowest absolute invariants. Corollary 4.3 Two absolute invariants for Abel’s equation in rational normal form case ii) are 1  A0 3 K(A, B) = 3B − (4.51) and A A 1  A0 h A0 0  A0  A0 i J(A, B) = 3B − 3B − + 5B − 2 3B − . A A A A A (4.52) For the rational normal form in case i) J and K are not defined. Proof For case ii), a3 = A, a2 = 0, a1 = B and a0 = 1 have to be substituted into (4.46) with the result Φ3 = 3A2 , and into (4.47) with the result  0  Φ5 = 3A3 3B − A and A h    i 0 0 A0 A0 Φ7 = 3A4 3B − A + 5B − 2 3B − . A A A Φ35 is the above invariant K(A, B) up to an irrelevant factor 9 which Φ53 7 is J(A, B). For the is due to a different normalization. The quotient Φ5 Φ Φ43 rational normal form of case i), a2 = a0 = 0 leads to Φ3 = Φ5 = 0, i.e., J and K are not defined. The ratio

The absolute invariants (4.51) and (4.52) are the decisive tool for solving equivalence problems of case ii). Theorem 4.16 arately.

The two rational normal forms have to be considered sep-

i) Any two Abel’s equations in rational normal form of first kind y 0 + A(x)y 3 + B(x)y = 0 and v 0 + P (u)v 3 + Q(u)v = 0 are equivalent to each other w.r.t. the group x = F (u), y = G(u)v.

184 ii) Any two equations in rational normal form of second kind y 0 + A(x)y 3 + B(x)y + 1 = 0 and v 0 + P (u)v 3 + Q(u)v + 1 = 0 are equivalent to each other w.r.t. the group x = F (u), y = F 0 (u)v if and only if for the absolut invariants (4.51) and (4.52) there holds K(A, B)|x=F (u) = K(P, Q)

(4.53)

and [3J(P, Q) − 5K(P, Q)]3 [3J(A, B) − 5K(A, B)]3 . = K(A, B) K(P, Q) x=F (u)

(4.54)

Proof In case i) the first equation is transformed by x = F (u) and y = G(u)v with the result v 0 + A(F )F 0 G2 v 3 +

h

B(F )F 0 +

G0 i v=0 G

with 0 = d . The conditions for equivalence are therefore du A(F )F 0 G2 = P,

B(F )F 0 +

G0 =Q G

or F0 =

P , A(F )G2

G0 = QG −

B(F ) P . A(F ) G

This first order system for the two undetermined functions F and G always has a solution depending on two constants. In case ii) the first equation is transformed by x = F (u) and y = F 0 (u)v. This leads to v 0 + A(F )F 03 v 3 +

h

B(F )F 0 +

F 00 i v+1=0 F0

from which the conditions for equivalence A(F )F 03 = P,

B(F )F 0 +

F 00 =Q F0

(4.55)

follow. This is a system of ode’s for the unknown function F . It may be transformed into a Janet basis by two reductions with the result 1  A0 3 1  P 0 3 3B − = 3Q − , A A P P x=F (u)

Equivalence and Invariants

1

h

A2/3

3B −

185

A0 i A0 0 1 A0  = − 3B − A 3A A x=F (u) 1 P 2/3

h

3Q −

Q0 i P 0 0 1 P 0  3Q − . − P 3P Q

Expressing these equations in terms of the invariants K and J, the conditions (4.53) and (4.54) are obtained. Up to now the following results have been obtained. The totality of Abel’s equations (4.45) comprises the following equivalence classes w.r.t. the group (4.41). Bernoulli’s equations form a single class, all other classes are parametrized by two rational functions, the absolute invariants (4.51) and (4.52). Example 4.14 invariants

Consider the Abel equations y 0 + xy 3 + y + 1 = 0 with


3 K(A, B) = 274 x − 13 and J(A, B) = 454 x2 − x x

11 15 x

+

1 5




x−

1 3




,

1 v + 1 = 0 with invariants and v 0 − 17 v 3 − 2u + u u2 K(P, Q) = −u(u − 3)3 and J(P, Q) = −3u u2 −

11 3 u


+ 5 (u − 3)

in the notation of the above theorem. The constraints (4.53) and (4.54) lead to u(u − 3)3 F 4 + 27F 3 − 27F 2 + 9F − 1 = 0, u5 (4u − 3)3 F 8 + 27F 3 − 108F 2 + 144F − 64 = 0. The greatest common divisor of its left hand sides is uF −1; it yields the trans1 . Consequently, the two equations are equivalent, formation function F = u 1 and y = − v . the transformation is achieved by x = u u2 Example 4.15 In the preceding example the first equation is modified to y 0 + xy 3 + 2y + 1 = 0 with the absolute invariants
3 K(A, B) = 216 x − 16 and J(A, B) = 360 x2 − x4 x4

11 30 x

+

1 20




x−

1 6




.

The second equation remains unchanged. The constraints (4.53) and (4.54) lead to u(u − 3)3 F 4 + 216F 3 − 108F 2 + 18F − 1 = 0, u5 (4u − 3)3 F 8 + 216F 3 − 432F 2 + 288F − 64 = 0. The gcd of the left hand sides is 1 now. Consequently, the two equations are not equivalent.

186

4.3

Nonlinear Equations of Second and Higher Order

For nonlinear equations of any order the results are far less complete than for linear or first order equations. They are basically limited to quasilinear equations of order two and three, and equations that are homogeneous in the dependent variable and its derivatives. Lie’s Equation. An important class of second order equations is y 00 + Ay 03 + By 02 + Cy 0 + D = 0

(4.56)

where A ≡ A(x, y), . . . , D ≡ D(x, y); almost all equations in Chapter 6 of Kamke’s collection are of this type. It is suggested that it is called Lie’s equation because Lie was the first to investigate it in detail in connection with his symmetry analysis. These aspects will be discussed in the subsequent Chapter 5. Most of the results in this subsection have been given for the first time by Roger Liouville in a series of articles between 1885 and 1900. Theorem 4.17 The structure invariance group of (4.56) is x = F (u, v), y = G(u, v) where v ≡ v(u), i.e., the full group of point-transformations of the plane. Proof From (B.7) it is obvious that the numerator of y 00 is linear in v 00 and a third order polynomial in v 0 , (B.6) shows that y 0 is a fraction with numerator and denominator linear in v 0 . Consequently, upon substitution into (4.56) an equation of the same structure is generated. A special case of this result is considered in Exercise 4.5. Theorem 4.18 (Liouville 1887) Lie’s equation (4.56) is equivalent to v 00 = 0 if and only if L1 = L2 = 0 where L1 ≡ Dyy + BDy − ADx + (By − 2Ax )D + 13 Bxx − 23 Cxy + 31 C(Bx − 2Cy ), L2 ≡ 2ADy + Ay D + 13 Cyy − 32 Bxy + Axx − 13 BCy + 23 BBx − Ax C − ACx . This important result singles out a single equivalence class of Lie’s equations. Later on in Theorem 5.11 it will turn out that the conditions L1 = L2 = 0 assure the maximal eight-parameter symmetry group for second order equations, i.e., they determine the symmetry class S82 . If they are not satisfied another set of invariants may be obtained. Theorem 4.19 (Liouville 1889) Lie’s equation allows a series of relative invariants w.r.t. point transformations which are determined by ν5 = L1 (L2 L1,y − L1 L2,y ) + L2 (L1 L2,x − L2 L1,x ) −AL31 + BL21 L2 − CL1 L22 + DL32 , νk+2 = L1 νk,y − L2 νk,x + kνk (L2,x − L1,y ),

k = 5, 7, . . .

(4.57)

Equivalence and Invariants

187

If x = φ(u, v), y = ψ(u, v) and ∆ ≡ ∂u φ∂v ψ − ∂v φ∂u ψ, these invariants are transformed according to νk (x, y)|x=φ,y=ψ = ∆k · νk (u, v). From these relative ν5 invariants the series of absolute invariants tk = kk is obtained. ν5 The invariants νk and tk are derived in Liouville [125], pages 19-23. Liouville describes a procedure of how to decide equivalence from the absolute invariants t7 and t9 . To this end, they must be defined and moreover they must be functionally independent. It turns out that this is not true for most equations that occur in actual problems, e.g. for the equations listed in Chapter 6 of Kamke’s collection. In these cases another series of invariants may be determined. Theorem 4.20 (Liouville 1889) If L1 L2 6= 0 and ν5 = 0, Lie’s equation allows the following series of invariants. At first the invariant w1 = 14 [L31 (α1 L1 − α2 L2 ) + R1 (L21 )x − L21 R1,x + L1 R1 ( 13 CL1 − DL2 )] or L1 w1 = 14 [L32 (α1 L2 − α0 L1 ) − R2 (L22 )y + L22 R2,y − L2 R2 (AL1 − 13 BL2 )] L2 is defined for L1 6= 0 or L2 6= 0 respectively. R1 , R2 , α0 , α1 and α2 are R1 = L1 L2,x − L2 L1,x + 13 BL21 − 32 CL1 L2 + DL22 , R2 = L1 L2,y − L2 L1,y + AL21 − 23 BL1 L2 + 13 CL22 , α0 = 13 By − Ax + 23 (AC − 13 B 2 ), α1 = 31 (Cy − Bx ) + AD − 19 BC, α2 = Dy − 13 Cx + 23 (BD − 13 C 2 ). From w1 the series of relative invariants wk+2 = L1 wk,y − L2 wk,x + k(L2,x − L1,y )wk is obtained from which the absolute invariants uk = wkk for k = 3, 5, . . . follow w1 if w1 6= 0. The invariants wk and uk are also derived in Liouville [125], pages 38-42. Finally, there is a third series of invariants if in addition there holds w1 = 0. Theorem 4.21 (Liouville 1889) If L1 6= 0, ν5 = 0 and w1 = 0, Lie’s equation allows the absolute invariant i2 =

3R1 + L2,x − L1,y L1

and the series of relative invariants i2k+2 = L1 i2k,y − L2 i2k,x + 2ki2k (L2,x − L1,y ), k ≥ 1.

188 if i2 6= 0. They yield the series of absolute invariants j2k = i2k ik2 These invariants may be found in Liouville [125], page 50. The case ν5 = 0, w1 = 0 deserves special attention because any Lie equation with this property is equivalent to a generalized Emden-Fowler equation. Theorem 4.22 (Babich and Bordag 1997) Any Lie equation (4.56) with ν5 = w1 = 0 is equivalent to an equation of the form v 00 + r(u, v) = 0. If in addition L2 = 0, the transformation to this form is determined by u = σ(x, y), v = ρ(x, y) with Z Z Z  Z  Z  σ(x, y) = exp 2 Bdy − Cdx dx and ρ(x, y) = exp Bdy dy. The proof may found in the article by Babich [8]. They discuss in detail the application of this result to the six types of Painlev´e’s equations. Example 4.16

Painlev´e’s equation III has the form y 00 =

δ y0 αy 2 + β y 02 − + + γy 3 + . y x x y

1 one obtains σ = log x, ρ = log y and the canonical With B = − y1 and C = x form v 00 = αeu+v + βeu−v + γe2(u+v) + δe2(u−v) . 3 y 02 − 3y 2 = 0 Equation 6.159 in Kamke’s collection y 00 − 4y 3 and fullfills the assumptions of the above theorem. Its coeffients B = − 4y 1 4 3 C = 0 yield σ = x, ρ = 256 y and the canonical form v 00 − 1024 v 5 = 0. It should be compared with Lie’s canonical form obtained in Example 6.9. Example 4.17

The Emden-Fowler canonical form obtained according to the above theorem is not unique as the next result shows. Theorem 4.23 (Babich and Bordag 1997) The structure invariance group of the Emden-Fowler equation y 00 + r(x, y) = 0 is Z x = c1 a(¯ x)2 d¯ x + c2 , y = a(¯ x)¯ y + b(¯ x) where a and b are undetermined functions of its argument, c1 and c2 are constants, and y¯ depends on x ¯. Quasilinear Equations of Second Order. In his award-winning article for the F¨ urstlich Jablonowski’sche Gesellschaft Tresse [181] (see also [180]) applied Lie’s theory of differential invariants to the equivalence problem of second order quasilinear ode’s of the form y 00 = ω(x, y, z) with z ≡ y 0 . In order

Equivalence and Invariants

189

to explain his results some notation has to be introduced first. If ψ ≡ ψ(x, y, z) is any function of x, y and z, define ψ10 =

∂ψ ∂ψ ∂kψ ∂ψ +z , ψ01 = and ψ k = . ∂x ∂y ∂y ∂z k

Due to the commutativity of the first two differential operators, the definition k ψij ≡ (ψ k )ij is meaningful. To any such symbol the two weights r = i − k + 2 and s = j + k − 1 are assigned. Furthermore, three differential parameters ∆x , ∆y and ∆z are defined in terms of their action on any invariant ψ with weights r and s. They return an invariant the order of which is increased by one. For ω 6= 0 they are defined as follows. ∆x ψ = ψ10 + ω∆z ψ+

h

(3r + 2s)



ω4 i 3ωω 5  +(2r + s) 10 ψ 4 5ω ω4

ω1 +

returns an invariant with weights r + 1 and s, 4 ω10 + ωω 5 + 2ω 1 ω 4 ω5 ∆ ψ + ∆z ψ x 5ω 4 ω4 n h ω2 4 4 o 3 ω 5 (ω10 + ωω 5 + 2ω 1 ω 4 ) i r + 2s ω01 − (3r + 2s) + + ψ 8 20 ω4 ω4 4 ω4 returns an invariant with weights r and s + 1 and

∆y ψ = ψ01 +

∆z ψ = ψ 1 + (r − s)

ω5 ψ 5ω 4

returns an invariant with weights r − 1 and s + 1. Applying this notation the invariants of y 00 = ω may be described as follows. Theorem 4.24 (Tresse 1896) A quasilinear ode y 00 = ω(x, y, z) with z ≡ y 0 and ∂ 4 ω/∂z 4 6= 0 has the following relative invariants w.r.t. point transformations x = σ(¯ x, y¯) and y = ρ(¯ x, y¯). i) Two invariants of order four: ω 4 with r = −2, s = 3 and 2 1 2 3 1 2 H ≡ ω20 − 4ω11 + 6ω02 − ω(3ω01 − 2ω10 ) + ω 1 (4ω01 − ω10 )

−3ω 2 ω01 + ω 3 ω10 + ωωω 4 with r = 2, s = 1. ii) Three invariants of order five: H10 ≡ ∆x H with r = 3, s = 1,

H01 ≡ ∆y H with r = s = 2,

K ≡ ∆z H with r = 1, s = 2. iii) Eleven invariants of order six: Ω6 ≡ ω 6 −

6 ω5 ω5 with r = −4, s = 5, 5 ω4

190

Ω510 ≡

5ω 4 (∆zx H − ∆xz H − ∆y H) with r = −2, s = 4, 24H

Ω501 = 94 (∆z Ω510 − ∆x Ω6 ) with r = −3, s = 5, Ω420 = ∆zz H −

Ω6 H with r = 0, s = 3, 5ω 4

Ω411 = 43 (∆z Ω420 − ∆x Ω510 ) with r = −1, s = 4, Ω402 ≡

4 Ω6 Ω420 Ω5 Ω5  ∆y Ω510 − ∆x Ω501 + + 10 410 with r = −2, s = 5, 4 5 ω 5ω ∆xz H with r = s = 2, ∆yz H with r = 1, s = 3, ∆xx H with r = 4, s = 1, ∆xy H with r = 3, s = 2, ∆yy H with r = 2, s = 3.

Any of these invariants I transforms according to I¯ (σx + y 0 σy )s−r = . I (σx ρy − σy ρx )s The proof involves rather lengthy calculations. They may be found in the above quoted article by Tresse. By inspection of the invariants it is seen that all invariants may be obtained by application of the differential parameters to ω 4 , H, Ω6 and Ω510 . The transformation law for the invariants given in Theorem 4.24 implies that absolute invariants may be formed from suitable powers of three properly chosen relative invariants. If ω 4 H 6= 0 and ψ is any relative invariant with weights r and s, an absolute invariant ψˆ =

ψ (ω 4 )α (H)β

is obtained if α = 18 (2s − r) and β = 18 (2s + 3r) are chosen. In order to obtain rational expressions, they are raised to the power of the least common multiple of the denominators of α and β. For the invariants up to order six the complete answer is given next. Absolute invariants are denoted by a hat on top of the corresponding symbol for the relative invariant. Corollary 4.4 With the same notation as in Theorem 4.24 and assuming ω 4 H 6= 0, the absolute invariants for y 00 = ω(x, y, z) of order 5 are ˆ = K

8 4 4 K8 ˆ 10 = (H10 ) ω , H ˆ 01 = (H01 ) . , H (ω 4 )3 (H)7 (H)11 ω 4 (H)5

Equivalence and Invariants

191

The invariants of order 6 are 5 4 5 8 6 4 ˆ 510 = (Ω10 ) , Ω ˆ 501 = (Ω01 ) , ˆ 6 = (Ω ) H , Ω Ω (ω 4 )7 (ω 4 )5 H (ω 4 )13 H

ˆ 420 = Ω

4 8 4 2 (Ω420 )4 ˆ 411 = (Ω11 ) , Ω ˆ 402 = (Ω02 ) , , Ω (ω 4 )3 (H)3 (ω 4 )9 (H)5 (ω 4 )3 H

(∆xz H)4 (∆yz H)8 (∆xx H)4 ω 4 (∆xy H)8 (∆yy H)2 , , , , . ω 4 (H)5 (ω 4 )5 (H)9 (H)7 ω 4 (H 13 ) ω 4 (H 3 ) Some of the results just described will be applied in Chapter 5.3 for determining symmetry classes of second order ode’s.

Exercises Exercise 4.1

Let a linear homogeneous ode be given in the form p0 y (n) + p1 y (n−1) + . . . + pn−1 y 0 + pn y = 0

with pk ≡ pk (x) polynomials in x for all k. Find a transformation that generates a new equation of the same type from it such that the constraint p1 = −p00 is valid. Discuss the result (Hirsch [72]). Exercise 4.2 Determine a transformation from the equation v 0 = 0 to a given equation y 0 + r(x, y) = 0. Compare the result with Theorem 4.10. Exercise 4.3 Derive the rational normal form obtained by choosing F (u) = u in (4.37). Exercise 4.4 Determine the structure invariance group of an Abel equation in Appel’s normal form y 0 + y 3 + r(x) = 0. Exercise 4.5 How does Lie’s equation (4.56) change if the dependent and the independent variable are exchanged by each other? Exercise √ 4.6 Show that the three equations y 00 + cy 0 + y = 0 and 1 0 y + 2 (3 ± i 3)cy + y = 0 are pairwise equivalent to each other. 00

Exercise 4.7 Show that the six equations y 00 − (a + 1)y 00 + ay 0 = 0 and 1, a 1 ,a 1 y 00 − (¯ a + 1)y 00 + a ¯y 0 = 0 with a ¯= a ¯ = 1 − a, a ¯ = 1− a ¯ = 1 − a and 1 are pairwise equivalent to each other. a ¯ =1+ a− 1 Exercise 4.8 Apply the algorithm LieSolve3.6 described on page 344 to equation (4.15). Why is it not effective for the particular application of Theorem 4.8 ?

192 Exercise 4.9 Design an algorithm that accepts a linear second order ode as input and returns a rational transformation to Weber’s equation (4.22) if there exists one or failed otherwise. Exercise 4.10

The same problem for the Bessel equation (4.16).

Chapter 5 Symmetries of Differential Equations

This chapter deals with the most important topic of this book, i.e., the nontrivial symmetries that a given ordinary differential equation (ode) may admit. Let two coordinate sets of the plane be defined with the additional assumption that in either of them one coordinate variable is dependent on the other. Roughly speaking a symmetry of an ode is a diffeomorphism connecting these coordinates for which this ode is an invariant. In the literature on symmetries these transformations are often called point transformations or variable transformations in order to distinguish them from more general transformations also involving the first derivative. This notation will frequently be used from now on whenever symmetries of an ode are the main topic. It is obvious that the entirety of symmetries of any given ode forms a group. The term symmetry group of a differential equation is applied to the largest group of transformations sharing this property. The Lie algebra of its infinitesimal generators forms the corresponding symmetry algebra. If a variable transformation is applied to a given differential equation, the symmetry group of the transformed equation is similar to the original one according to Definition 3.2. The equivalence class to which the symmetry group of a particular ode belongs is called its symmetry type. Consequently, all equations contained in an equivalence class have the same symmetry type. The reverse is not true. As a consequence, the entirety of all differential equations allowing the same type of symmetry group is the union of equivalence classes. Krause and Michel [96] called it the stratum of ode’s corresponding to a symmetry type. In this book it will be called the symmetry class. The symmetries of a differential equation occur in different connections. On the one hand, there is the classification problem. Its aim is to determine all possible symmetry types for a family of ode’s, e. g. ode’s of a fixed order. The starting point for this approach is the listing of groups given in Section 3.4; its differential invariants determine the general form of an ode that may be invariant under the respective group. On the other hand, if any particular ode is given, its symmetry type has to be determined if it is to be applied for finding its solutions. The symmetry problem for equations of order one is significantly different from that of equations of order two or higher. The main difference is the fact that for first order equations in general there is no algorithm available for determining any symmetry generator; only heuristics or insight into the

193

194 problem from which the equation originates may allow finding them. What makes the problem especially difficult is the fact that any first order ode allows infinitely many symmetries as discussed at the beginning of Section 5.2. Contrary to this, for equations of order two or higher, the symmetry type may always be identified algorithmically, and there is a well-defined procedure for transforming it to a canonical form. The key for this is the Janet basis for the determining system and the theorems derived in Section 3.5. In the subsequent Section 5.1, general properties of the behavior of differential equations under a change of variables are discussed and various concepts related to its symmetries are defined. Section 5.2 discusses the symmetry structure of first order equations. The most important field of application for Lie himself was equations of second order; they are the subject of Section 5.3. In Section 5.4 a complete discussion of equations of order three is given following a similar approach to the second order equations. Finally in Section 5.5 the symmetries of linear equations of any order are discussed.

5.1

Transformation of Differential Equations

Introducing new variables into a given ode has been a popular method in order to faciliate the solution procedure all the time. Usually this is done in an ad hoc manner without guaranteed success. In particular there is no criterion for deciding whether a certain class of transformations will lead to an integrable equation or not. A critical examination of these methods was the starting point for Lie’s symmetry analysis. Therefore the behavior of ode’s under various kinds of transformations will be investigated first. Properties of Point Transformations. Let an ode of order n in the independent variable x and the dependent variable y ≡ y(x) be given as ω(x, y, y 0 , . . . , y (n) ) = 0.

(5.1)

If not stated explicitly, it will be assumed that ω is a polynomial in the derivatives with coefficients in some base field which is usually the field of rational functions in x and y, i.e., ω ∈ Q(x, y)[y 0 , . . . , y (n) ]. Any other field that occurs later on during the solution procedure is an extension of this base field. A point-transformation between two planes with coordinates (x, y) and (u, v), and dependencies y ≡ y(x) and v ≡ v(u) respectively, is considered in the form u = σ(x, y), v = ρ(x, y). (5.2) The term point-transformation expresses the fact that only the coordinates occur as arguments of the transformation functions. If, in addition, the first

Symmetries of Differential Equations

195

derivative is transformed by an independent function of x, y and y 0 of a certain form it is called a contact-transformation. They are not dicussed in this book. In general equations (5.2) define a diffeomorphism between the two coordinate planes. Depending on the particular situation, the function field in which σ and ρ are contained has to be specified. Let a curve in the x − y-plane described by y = f (x) be transformed under df (5.2) into v = g(u). There arises the question of how the derivative y 0 = dx dg corresponds to v 0 = under this transformation. A simple calculation leads du to the first prolongation v0 =

ρx + ρy y 0 dv = ≡ χ1 (x, y, y 0 ). du σx + σy y 0

(5.3)

It is remarkable that the knowledge of (x, y, y 0 ) and the equations of the point transformation (5.2) are sufficient for computing the derivative v 0 , knowing the equation of the curve is not required. This is also expressed by saying that the line element (x, y, y 0 ) is transformed into the line element (u, v, v 0 ) under the action of a point transformation. Similarly the transformation law for derivatives of second order is obtained as v 00 =

dv 0 χ1,x + χ1,y y 0 + χ1,y0 y 00 = ≡ χ2 (x, y, y 0 , y 00 ). du σx + σy y 0

(5.4)

Explicitly in terms of σ and ρ it is v 00 =

1 {(σx ρy − σy ρx )y 00 + (σy ρyy − σyy ρy )y 03 (σx + σy y 0 )3 +[σx ρyy − σyy ρx + 2(σy ρxy − σxy ρy )]y 02

(5.5)

+[σy ρxx − σxx ρy + 2(σx ρxy − σxy ρx )]y 0 + σx ρxx − σxx ρx }. In general there holds v (n) = and v (n+1) =

dv (n−1) ≡ χn (x, y, y 0 , . . . , y (n) ) du

(5.6)

χn,x + χn,y y 0 + . . . + χn,y(n−1) y (n) dv (n) = . du σx + σy y 0

The form of a differential equation is extremely sensitive to a variable change. The following definition introduces a particular term for the exceptional cases where this is not true. Definition 5.1 (Symmetry of an ode) Equation (5.1) is said to be invariant under the transformation x = φ(u, v),

y = ψ(u, v) with v ≡ v(u)

(5.7)

196 if the functional dependence on u, v, v 0 , . . . , v (n) of the transformed equation is the same as that of the original equation (5.1) on x, y, y 0 , . . . , y (n) , i.e., if it is a differential invariant for (5.7). Such a transformation is called a symmetry of the differential equation. The general solution of equation (5.1) is a family of curves φ(x, y, C1 , . . . , Cn ) = 0

(5.8)

in the x−y-plane depending on n constant parameters C1 , . . . , Cn . The transformation (5.7) acts on the curves (5.8) just as on the differential equation. If it is a symmetry, the functional dependence of the transformed curves on u and v is the same as the dependence on x and y in (5.8). This is not necessarily true for the parameters because they do not occur in the differential equation itself. This means, the entirety of curves described by the two equations is the same; to any fixed values of the constants, however, there may correspond a different curve in either set. In other words, the solution curves are permuted among themselves by a symmetry transformation. Example 5.1 Consider the ode 2yy 00 − y 02 = 0 with the general solution a1 v a1 u + a , y = y = (C1 x + C2 )2 . Its symmetry group is x = 1 − 3 a2 u (1 − a2 u)2 where a1 , a2 and a3 are the group parameters. In the transformed variables the solution is v = (C¯1 u + C¯2 )2 with 3 + 1C . C¯1 = √1 [C1 (a1 − a2 a3 ) − C2 a2 ], C¯2 = a√ a1 a1 2

From the above Definition 5.1 of a symmetry of an ode and the geometric considerations on its solution curves it is obvious that all symmetry transformations of a given ode form a group. This important concept is introduced next. Definition 5.2 (Symmetry group, symmetry type) The totality of symmetry transformations of an ode forms a continuous group; it is called the symmetry group of that equation. The type of this group is called its symmetry type. At this point the connection to Chapter 3 of this book becomes obvious. All that has been said about groups of transformations of a two-dimensional manifold applies to the symmetry groups of ode’s. In particular this applies to the classification of these groups and the resulting constraints for the possible symmetries of a differential equation. In order to establish a systematic theory of symmetries of ode’s, it must be known how they behave under variable transformations. The answer is given in the following theorem. Theorem 5.1 (Lie 1891) Equivalent equations have the same symmetry type. The reverse is not true. In general to any symmetry type, there corresponds the union of one or more equivalence classes.

Symmetries of Differential Equations

197

Proof Let ω(x, y, y 0 , . . . , y (n) ) = 0 be a differential equation in x and y ≡ y(x), and ω ¯ (¯ x, y¯, y¯0 , . . . , y¯(n) ) = 0 the transformed equation under the variable change Φ : x = α(¯ x, y¯), y = β(¯ x, y¯) with y¯ ≡ y¯(¯ x) and inverse Φ−1 . Furthermore, let g : x = φ(u, v), y = ψ(u, v) with v ≡ v(u) be a symmetry transformation of ω with the result ω(u, v, v 0 , . . . , v (n) ) = 0. Then g¯ ≡ Φ−1 gΦ transforms ω ¯ (¯ x, y¯, y¯0 , . . . , y¯(n) ) = 0 into ω ¯ (¯ u, v¯, v¯0 , . . . , v¯(n) ) = 0, i.e., it defines a symmetry of ω ¯ of the same type as g. On the other hand, by Theorems 6.4 and 6.16, the equations y 00 +y 04 +1 = 0 2 3 and y 000 + y 00 y 0 + 1 both belong to symmetry classes S2,1 and S2,1 respectively with the same symmetry type, yet due to its different order they are obviously not equivalent to each other. In order to utilize the symmetries of an ode for the solution procedure, the geometrical considerations above must be expressed in analytical terms. Of particular importance is the fact that this may be achieved in terms of the infinitesimal generators of a symmetry, the expressions for its finite transformations are not required. Due to its importance for the rest of this book the prolongation of a vector field in x and y(x) is introduced next. It is a specialization of Definition 3.4 in Chapter 3. Definition 5.3 (Prolongation in the x-y-plane) Let an infinitesimal generator U = ξ(x, y)∂x + η(x, y)∂y be given and y ≡ y(x) depend on x. Its nth prolongation is U (n) = U + ζ (1)

∂ ∂ ∂ + ζ (2) 00 + . . . + ζ (n) (n) . ∂y 0 ∂y ∂y

(5.9)

The functions ζ (k) are recursively defined by ζ (1) = D(η) − y 0 D(ξ),

ζ (k) = D(ζ (k−1) ) − y (k) D(ξ)

(5.10)

for k ≥ 2, D = ∂x + y 0 ∂y + y 00 ∂y0 . . . is the operator of total differentiation with respect to x. From (5.9) it is obvious that the prolonged generators operate on the space of derivatives up to order n. A few remarkable properties of the functions ζ (k) are formulated subsequently as a lemma. They follow immediately from its definition and are quite useful for practical calculations. Lemma 5.1 erties.

The functions ζ (k) defined by (5.10) have the following prop-

1. They are linear and homogeneous in ξ(x, y), η(x, y) and its derivatives up to order k. 2. Its coefficients depend explicitly on y 0 , y 00 , . . . , y (k) ; for k > 1, y (k) occurs linearly and y 0 with power k+1. In particular they do not depend explicitly on x or y.

198 From this lemma it is obvious that the amount of computations grows enormously with the order of the differential equation due to the increase in the number of terms in ζ (k) . A rough estimate for this number is 2k . For k = 1, . . . , 10 the exact figure is given in the subsequent table. k 1 2 3 N umber of terms 4 9 17

4 5 6 7 8 9 10 29 47 73 110 162 234 332

The three lowest ζ 0 s are 0

ζ (1) = ηx + (ηy − ξx )y 0 − ξy y 2 , 0

ζ (2) = ηxx + (2ηxy − ξxx )y 0 + (ηyy − 2ξxy )y 2 − ξyy y

0

3

+(ηy − 2ξx )y 00 − 3ξy y 0 y 00 , 0

ζ (3) = ηxxx + (3ηxxy − ξxxx )y 0 + 3(ηxyy − ξxxy )y 2 + (ηyyy − 3ξxyy )y 0

0

3

0

−ξyyy y 4 + 3(ηxy − ξxx )y 00 + 3(ηyy − 3ξxy )y 0 y 00 − 6ξyy y 2 y 00 +(ηy − 3ξx )y 000 − 4ξy y 0 y 000 − 3ξy y 002 . (5.11) It is convenient to have the prolongations for some simple vector fields explicitly available. For linear coefficients ξ = ax, η = by or ξ = ay, η = bx, a and b constant, the third prolongations are U (3) = ax∂x + by∂y + (b − a)y 0 ∂y0 + (b − 2a)y 00 ∂y00 + (b − 3a)y 000 ∂y000 U (3) = ay∂x + bx∂y + (b − ay 02 )∂y0 − 3ay 0 y 00 ∂y00 − a(4y 0 y 000 + 3y 002 )∂y000 . For ξ = x2 , η = y 2 it is U (3) = x2 ∂x + y 2 ∂y + 2(y − x)y 0 ∂y0 + 2[y 0 (y 0 − 1) + (y − 2x)y 00 ]∂y00 +2[3y 00 (y 0 − 1) + (y − 3x)y 000 ]∂y000 . For later use a relation between applying the prolongation operator and taking the total derivative is given next. Lemma 5.2 The prolongation (5.9) of U = ξ∂x + η∂y and the operator of total differentiation D obey U (n+1) D(φ) = D(U (n) φ) − D(ξ)D(φ); φ is an undetermined function of x, y(x) and its derivatives up to any order. The proof is based on expanding the left and the right hand side of this relation; it is left as Exercise 5.1. The next result which is due to Lie [113], Kapitel 16, is fundamental for the further proceeding because it is the basis of a constructive method for determining the symmetries of large classes of ode’s. It is an immediate consequence of Definition 5.2.

Symmetries of Differential Equations

199

Theorem 5.2 (Lie 1891) A differential equation ω(x, y, y 0 , . . . , y (n) ) = 0 with n ≥ 1 is invariant under a transformation with infinitesimal generator U = ξ(x, y)∂x + η(x, y)∂y if and only if U (n) ω(x, y, y 0 , . . . , y (n) ) = 0 mod (ω = 0)

(5.12)

in the space of variables x, y, y 0 , . . . , y (n) . There is a remarkable constraint for the number of possible symmetries of an ode if the order is not less than two which is also due to Lie. In addition to the theoretical interest in this result it has the important consequence that the symmetry classification becomes a finite problem. Theorem 5.3 (Lie 1893) The maximal number of symmetry generators of an ode of order n ≥ 2 is finite. For n = 2 it is 8, for n ≥ 3 it is n + 4. The proof which is based some geometric considerations may be found in Lie [114], Kapitel 12, Satz 3. For reasons that will become clear shortly the further discussion is limited to n ≥ 2. The case n = 1 will be considered in Section 5.2. In order to make the symmetry problem for ode’s manageable the functional dependence of ω on its arguments is restricted as follows. If not stated otherwise it is assumed to be quasilinear, i.e., linear in the highest derivative, and rational in its remaining arguments. Consequently, equations of the form y (n) + r(x, y, y 0 , . . . , y (n−1) ) = 0

(5.13)

are considered with r ∈ Q(x, y, y 0 , . . . , y (n−1) ). This class covers a wide range of interesting equations as it may be seen from the respective chapters of the collection by Kamke [85]. The quasilinearity and the rationality in the derivatives have another important property, they are invariant under general, i.e., not necessarily rational point transformations as it may be seen from eqs. (5.3) to (5.4). Based on Theorem 5.2 a procedure for determining the infinitesimal symmetry generators of an ode (5.13) may be designed as follows. Due to the assumptions on the form of ω, the expression at the left hand side of (5.12) is rational in the derivatives with coefficients depending on x and y, the unknown functions ξ(x, y) and η(x, y) and derivatives thereof. Because the dependence on the derivatives y (k) is completely explicit, it can only vanish if each coefficient of the monomials in the derivatives occuring in the numerator vanishes. As a consequence of Lemma 5.1, the resulting constraints form a system of linear homogeneous partial differential equations for ξ and η in which derivatives y (k) do not occur. It is called the determining system. Its general solution defines the symmetries of the original differential equation. The algorithms described in Chapter 2 are applied in the following algorithm for determining large classes of symmetry generators.

200 Algorithm 5.1 Symmetries(ω). Given a quasilinear ode ω of order n ≥ 2, its infinitesimal symmetry generators with Liouvillian coefficients are returned. S1 : Set up determining system. Reduce the n−th prolongation of ω w.r.t. ω, collect the coefficients of the monomials in the derivatives and equate them to zero. S2 : Janet basis. By means of the algorithm given in Chapter 2 transform the system obtained in S1 into a Janet basis. S3 : Determine coefficients. Decompose the Janet basis obtained in S2 applying the algorithms of Chapter 2 and determine the Liouvillian solutions from it. If there is no nontrivial solution, return failed. S4 : Return vector fields. From the solutions obtained in S3 construct the infinitesimal generators and return them. This algorithm will be applied frequently in later parts of this chapter, both for determining the symmetry generators explicitly, or the symmetry class to which a given equation belongs. It should be emphasized, however, that symmetry generators with non Liouvillian coefficients in general cannot be obtained from it. Example 5.2 Equation 6.159 of Kamke’s collection 4y 00 y − 3y 02 − 12y 3 = 0 generates the determining system 3 ξ = 0, η + 3y 2 η − 6y 2 ξ − 6yη = 0, ξyy + 4y y xx y x 3 η − 9 y 2 ξ = 0, η − 2ξ − 3 η + 3 η = 0. ηxy − 12 ξxx − 4y x y yy xy 2 4y y 4y 2 This is the result of step S1 of the above algorithm. The Janet basis ξx +

1 1 η = 0, ξy = 0, ηx = 0, ηy − η = 0 2y y

is obtained in step S2 from which the solution ξ = − 12 C1 x + C2 , η = C1 y in step S3 follows. Finally in step S4 the two symmetry generators U1 = ∂x and U2 = x∂x − 2y∂y are obtained. Its commutator [U1 , U2 ] = U1 shows that they are canonical generators of the group g25 . Classification of Differential Invariants. In order to obtain a classification of possible symmetry types of ode’s, the proper differential invariants of all finite transformation groups of a two-dimensional manifold have to be determined. The starting point for this classification is the listing of transformation groups as described in Chapter 3. Coordinates are x and y, y depends on x. For an r-parameter group with infinitesimal generators ξi p + ηi q for i = 1, . . . , r, the differential invariants of order m are solutions of the linear

Symmetries of Differential Equations

201

homogeneous system (1) (m) ∂Φ = 0, ξ1 ∂Φ + η1 ∂Φ + ζ1 ∂Φ0 + . . . + ζ1 ∂x ∂y ∂y ∂y (m) (1) (m) ∂Φ ξ2 ∂Φ + η2 ∂Φ + ζ2 ∂Φ0 + . . . + ζ2 = 0, ∂x ∂y ∂y ∂y (m) .. .. . . (m) ∂Φ (1) = 0. ξr ∂Φ + ηr ∂Φ + ζr ∂Φ0 + . . . + ζr ∂x ∂y ∂y ∂y (m)

(5.14)

In this subsection, all systems of pde’s are represented in grlex term order with x > y > y 0 > y 00 > y 000 . The group property guarantees that this is a complete system for Φ with m + 2 − r solutions for 1 ≤ r ≤ m + 2. Due to Lemma 5.1 their dependencies may be chosen such that Φ1 ≡ Φ1 (x, y, y 0 , . . . , y (r−1) ), Φ2 ≡ Φ2 (x, y, y 0 , . . . , y (r) ), .. .. . . Φm−r+2 ≡ Φm−r+2 (x, y, y 0 , . . . , y (m) ). The two lowest invariants Φ1 and Φ2 are called fundamental invariants. They are determined by the methods described in Section 3.2 for solving Jacobian systems. The Jacobian scheme breaks down if the determinant of the coefficient matrix of the first r − 2 rows vanishes. Correspondingly the lower equations of the group are obtained from the irreducible factors of the Lie determinant ξ η ζ (1) . . . ζ (r−2) 1 1 1 1 ξ2 η2 ζ2(1) . . . ζ2(r−2) (5.15) ∆= . . . .. . .. .. .. . ξr ηr ζr(1) . . . ζr(r−2) Its irreducible factors may determine additional invariants. According to Lie [109], part I, page 247, the higher invariants may be obtained by differentiation as it is shown next. Lie gave some intuitive geometric arguments in favor of this result; see also the remarks by Krause and Michel [96] on page 254, footnote 4. This important result is proved next. The higher invariants Φj for j ≥ 3 may be obtained by dΦj . differentiation Φj+1 = dΦ1 Theorem 5.4

DΦ The above relation may be written as Φj+1 = DΦj because by 1 dΦj definition there holds = DΦj and dΦ1 = DΦ1 . Let Φj be an invariant of dx dx order n. If Φj+1 is an invariant of order n + 1, it has to obey U (n+1) Φj+1 = 0. Proof

202 The numerator of the left hand side is D(Φ1 )U (n+1) D(Φj ) − D(Φj )U (n+1) D(Φ1 ) = D(Φ1 )[D(U (n) Φj ) − D(ξ)D(Φj )] − D(Φj )[D(U (n) Φ1 ) + D(ξ)D(φ1 )] = 0. In the last line Lemma 5.2 has been applied. Due to the assumption that Φ1 and Φj are invariants there holds U (n) Φ1 = 0 and U (n) Φj = 0. The remaining terms cancel each other. Example 5.3 Extending the three generators of the group g13 defined on page 139 up to third order yields the following system of pde’s for any invariant Φ. Φx = 0,

2xΦx + yΦy − y 0 Φy0 − 3y 00 Φy00 − 5y 000 Φy000 = 0,

x2 Φx + xyΦy − (y 0 x − y)Φy0 − 3y 00 xΦy00 − (5y 000 x + 3y 00 )Φy000 = 0 with ∆ = y 2 . Applying the algorithm JanetBasis of the previous chapter generates the Jacobian form X1 ≡ Φx = 0,

3y 00 X2 ≡ Φy0 − y Φy000 = 0,

3y 00 5y 000 y + 3y 00 y 0 000 X3 ≡ Φy − y Φy00 − Φy = 0. y2 The first equation implies that the invariants do not depend explicitly on x. In the last equation, the third derivative y 000 does not occur in the coefficients of Φy and Φy00 . Therefore a solution independent of y 000 may be obtained dy dy 00 from 3 y + 00 = 0 with the result Φ1 = y 00 y 3 . Because also X1 Φ1 = 0 and y X2 Φ1 = 0, Φ1 is the first invariant searched for. dy 000 dy 0 For the second equation a solution φ is obtained from 3 y + 00 = 0 y φ 000 00 0 in the form φ = y y + 3y y . Because X3 φ = −4 y , the extension of φ to a solution of the full system must have the form Φ(φ, y) and satisfies the 4φ dφ dy equation ∂Φ y + ∂Φ = 0 or equivalently + 4 y = 0, i.e., φy 4 = C. ∂φ ∂y φ Therefore Φ = φy 4 is a solution of the complete system, the second invariant is Φ2 = y 000 y 5 + 3y 00 y 0 y 4 . Example 5.4 Consider the group g25 with P generators ∂y and x∂x + y∂y . n The pde’s for invariants up to order n are xΦx − k=2 (k −1)y (k) Φy(k) = 0 and 0 k−1 (k) Φy = 0, with invariants Φ1 = y , Φk = x y for k = 2, . . . , n. Applying dΦ Φ dΦ 2Φ 2 3 3 3 + Φ4 whereas the above Theorem 5.4 yields = 1 + Φ and = Φ2 dΦ1 dΦ1 2 formula leads to Φ2 = xy 00 , Φ3 = x2 y 000 and Φ4 = x3 y (4) . It shows that the invariants obtained by differentiation may be not in simplest form.

Symmetries of Differential Equations

203

The listing below contains the complete information on the possible groups of point symmetries of ordinary differential equations of arbitrary order. It is due to Lie [109], part I. Some minor corrections have been included that may be found in the Anmerkungen by Engel in vol. V of the Gesammelte Abhandlungen, pages 676-679. Furthermore, various corrections and improvements described in the dissertation of Heineck [69] have been added. In general the necessary calculations proceed similarly to the above examples. (k) (k) In the subsequent tabulation the fundamental invariants Φ1 and Φ2 , (k) and the Lie determinants ∆ for the groups gk , k = 1, . . . , 27, are explicitly given. Different from Lie’s listing, all invariants are rational. This faciliates the explicit calculations that will be performed later on. In various cases the Schwarzian derivative of y is denoted by the abbreviation  00 2 y y 000 (5.16) w = 0 − 32 0 . y y [3y 00 y (4) − 5y 0002 ]3 (1) 9y 002 y (5) − 45y 00 y 000 y (4) + 40y 0003 , Φ2 = , 008 y 3y 004 ∆(1) = 9y 003 . [9y 002 y (5) − 45y 00 y 000 y (4) + 40y 0003 ]2 (2) Group g2 : Φ1 = , 3(3y 00 y (4) − 5y 0002 )3 (1)

Group g1 : Φ1 =

(2)

Φ2 =

9y 003 y (6) − 63y 002 y 000 y (5) + 105y 00 y 0002 y (4) − 35y 0004 , 3(3y 00 y (4) − 5y 0002 )2

∆(2) = 2y 002 (5y 0002 − 3y 00 y (4) ). 3 (3) (3) Group g3 : Φ1 = u8 , Φ2 = v3 where ρ5 ρ5 ρ4 = 3y 00 y (4) − 4y 0003 ,

ρ5 = 3y 002 y (5) − 15y 00 y 000 y (4) +

ρ6 = 3y 002 y (6) − 24y 002 y 000 y (5) + 60y 00 y 0002 y (4) − 40y 004 (7)

ρ7 = 9y y

003 000 (6)

− 105y y y

002 0002 (5)

+ 420y y

y

40 0003 , 3 y 0004

,

00 0003 (4)

− 7000y y

y

+

ρ8 = 27y 005 y (8) − 48y 000 ρ7 − 840y 0002 ρ6 − 2240y 0003 ρ5 − 2800y 0004 ρ4 −

1120 0005 , 3 y 2240 0006 , 3 y

u = 2ρ5 ρ7 − 35ρ4 ρ25 − 7(ρ6 − 35 ρ24 )2 , v = ρ5 (ρ8 − 84ρ4 ρ6 +

245 3 3 ρ4 )

− 12(ρ7 −

35 2 ρ4 ρ5 )(ρ6

5 2 3 − 53 ρ24 ) + 28 ρ5 (ρ6 − 3 ρ4 ) ,

∆(3) = −2y 00 (9y 002 y (5) − 45y 00 y 000 y (4) + 40y 0003 )2 . (4)

(4)

Group g4 :

Φ1 = x, Φ2 =

Group g5 :

Φ1 =

Group g6 :

Φ1 =

(5)

(6)

(5) y 00 , Φ2 y0

(4) y 00 y 000 , ∆(4) = 0. 0 , Φ3 = y y0 y 000 = 0 , ∆(5) = y 0 . y

(6) y 0 y 000 y 02 y (4) , ∆(6) = −y 0 y 00 . 002 , Φ2 = y y 003

204 (7)

(7)

Φ1 = y 00 y 0(2−c)(c−1) , Φ2 = y 000 y 0(3−c)(c−1) , (7) ∆ = (c − 1)y 0 , c 6= 1.

Group g7 :

(8)

(8)

Φ1 = x, Φ2 = w, ∆(8) = 0.

Group g8 :

(9)

(9)

Group g9 : Φ1 = w, Φ2 = w0 , ∆(9) = 2y 03 . (10)

Group g10 : Φ1

(10)

Φ2

=

∆(10)

[(x − y)y 00 + 2y 0 (y 0 + 1)]2 , y 03

=

(x − y)2 y 000 + 6(x − y)(y 0 + 1)y 00 6(y 02 + 4y 0 + 1) , + 02 y0 y = 2(y − x)2 y 0 .

02 00 (11) = w 3 , Φ2 = w 2 , ∆(11) = 4y 02 (y 0 y 000 − 32 y 002 ). w w 00 02 (4w2 w000 − 18ww0 w00 + 15w03 )2 (12) (12) 4ww − 5w Group g12 : Φ1 = , Φ2 = , 3 w w9 3 002 (12) 02 0 000 ∆ = 4y (y y − 2 y ). (11)

Group g11 : Φ1

(13)

Group g13 :

Φ1

∆(14)

= y 000 y 5 + 3y 00 y 0 y 4 , ∆(13) = 2y 2 .

(yy 000 + 3y 0 y 00 )2 (14) 3yy 00 y (4) − 4yy 0002 , Φ2 = , 003 yy y 003 = 2y 2 y 00 .

(14)

Group g14 : Φ1

(13)

= y 00 y 3 , Φ2

=

Group g15 : Let the determinants D and Di , for i = 1, 2, . . . be defined by φ 1 . . D = . φr y (15)

Then Φ1 Group Group

(r)

. . . φ1 .. .

(15)

(r−1) (r+i) . . . φ1 φ1 .. .. . . . (r−1) (r+i) φ0r . . . φr φr y 0 . . . y (r−1) y (r+i)

φ01 .. .

(15)

= D1 , ∆(15) = 0. 1 = (log D)0 = D = x, ∆(16) = 0. D with D as above, 0 (17) (17) = c0 y + c1 y 0 + . . . + cr y (r) , Φ2 = Φ1 ,

= x, Φ2

(16) g16 : Φ1 (17) g17 : Φ1

φ 1 . . Di = . φr y

, (r) φ0r . . . φr y 0 . . . y (r)

φ01 .. .

= D, Φ3

(16) Φ2

∆(17)

φ1 = ... φr

. (r−1) 0 φr . . . φr

φ01 .. .

(r−1)

. . . φ1 .. .

Group g18 : In terms of the previously defined determinants D, D1 and D2 they may be written as (18)

Φ1

=

D1 , D

(18)

Φ2

=

D2 D

Symmetries of Differential Equations

205

if the proper P values for the φk are substituted. They are of order l + and l + ρk + 2 respectively, ∆(18) = D. Group g19 : For c arbitrary: ∆(19) = (c − r)y (r) , c−r

(19)

for c 6= r: Φ1

(19)

for c = r: Φ1

(20)

Group g20 : Φ1

(21)

Group g21 : Φ1

(22)

Group g22 : Φ1

P

ρk + 1

c−r

(19) y (r+1) y (r+2) , Φ2 = c−r−2 , (r) c−r−1 y y (r) (19) y (r+2) = y (r) , Φ2 = 2. y (r+1)

=

= y (r+1) ey

(r)

/r!

(20)

, Φ2

= y (r+2) e2y

(r)

/r!

, ∆(20) =

Qr

k=1

k!.

(r) 2 (r+3)

(21) y (r) y (r+2) y y (21) = = y (r) y (r+1) . 2 , Φ2 3 , ∆ y (r+1) y (r+1) 2 v1r+1 v2r+1 (22) , Φ2 = , ∆(22) = y (r) = 2(r+3) 3(r+3) y (r) y (r)

=

where 2

v1 = (r + 1)y (r) y (r+2) − (r + 2)y (r+1) , 2

v2 = (r + 1)2 y (r) y (r+3) − 3(r + 1)(r + 3)y (r) y (r+1) y (r+2) 3

+2(r + 2)(r + 3)y (r+1) . v22 (23) , Φ2 = v32 v13 v1 where v1 and v2 are the same as for the preceding group and (23)

Group g23 : Φ1

=

3

2

v3 = (r + 1)3 y (r) y (r+4) − 4(r + 1)2 (r + 4)y (r) y (r+1) y (r+3) 2

4

+6(r + 1)(r + 3)(r + 4)y (r) y (r+1) y (r+2) − 3(r + 2)(r + 3)(r + 4)y (r+1) ,   2 ∆(23) = y (r) (r + 2)y (r+1) − (r + 1)y (r) y (r+2) . y 000 , ∆(24) = 0. y 002

(24)

= y 0 , Φ2

(25)

= y 0 , Φ2

(26)

= y (k) for k ≥ 1, ∆(26) = 1.

(27)

= x, Φk

Group g24 : Φ1 Group g25 : Φ1

Group g26 : Φk Group g27 : Φ1

(24)

=

(25)

= xy 00 , Φ3

(27)

(25)

= x2 y 000 , ∆(25) = y.

= y (k) for k ≥ 1, ∆(27) not defined.

There is an important distinction on whether an invariant equation originates from the invariant of a group or from a lower equation. In the latter case, the left hand side of an invariant equation in normal form is identical to the factor of the lower equation as given in the above listing. Invariant equations originating from an invariant may be any function of them.

206

5.2

Symmetries of First Order Equations

Although first order quasilinear equations ω ≡ y 0 + r(x, y) = 0 are simpler than their higher order counterparts, applying its symmetries for solving them is limited by the fact that there is no algorithmic procedure for obtaining the symmetry generators explicitly. This is obvious from the expression at the left hand side of (5.12) for n = 1; it is a single linear homogeneous pde of first order for the two unknown functions ξ(x, y) and η(x, y). With ω as given above it is ξrx + ηry + ηx − (ηy − ξx )r − ξy r2 = 0. (5.17) There is no solution algorithm available for this pde. What makes things even worse, there are a lot of trivial solutions that cannot be utilized for the solution procedure as it will turn out later on. They yield the trivial symmetries ξ = φ(x, y) and η = −rφ(x, y) with φ(x, y) an undetermined function of its arguments. The problem is to find the nontrivial solutions that may be applied for the solution procedure. Classification of Symmetries. According to Theorem 4.10 the quasilinear first oder equations form a single equivalence class for which v 0 = 0 is a canonical form. Consequently, there is only a single symmetry class, i.e. all equations of this kind share the same symmetries up to similarity. They are described in the subsequent theorem. Theorem 5.5 Any symmetry generator of a first order quasilinear ode v’=0 in canonical variables u and v ≡ v(u) is an operator α(u, v)∂u + β(v)∂v of the infinite-dimensional Lie algebra corresponding to the transformation group S 1 ≡ G with Janet basis {βu }; α and β undetermined functions of its arguments. Proof The determining system for the coefficients α(u, v) and β(u, v) in canonical variables is immediately obtained from (5.12) in the form βu = 0 with the solution α(u, v) and β(v). This theorem describes the symmetries of first order ode’s in full generality. Any finite group of invariance transformations of such an equation must be up to similarity contained in the pseudogroup given there. From the listing of groups and invariants in the preceding section it follows that the group g12 with generators ∂u , u∂u , u2 ∂u , ∂v , v∂v , v 2 ∂v leaves v 0 = 0 invariant. An arbitrary first order equation y 0 + r(x, y) = 0 not in canonical variables allows the same invariance transformations; its symmetry generators are similar to those just mentioned. Corollary 5.1 Any symmetry generator of a first order quasilinear ode y 0 + r(x, y) = 0 obtained from v 0 = 0 by the transformation u = σ(x, y), v = ρ(x, y) with the inverse x = φ(u, v), y = ψ(u, v) has the form [α(u, v)(φu ∂x + ψu ∂y ) + β(v)(φv ∂x + ψv ∂y )]u=σ(x,y),v=ρ(x,y) with α(u, v) and β(v) undetermined functions of its arguments.

Symmetries of Differential Equations

207

This is a straightforward consequence of Theorem 5.5 and the transformation formulas of Lemma 3.3. The above corollary may be applied to obtain finite groups of invariance transformations for first order ode’s as the following example shows. For the Riccati equation of Example 4.10 the transformation 1 and ρ = x(xy + 1) with the functions to canonical variables were σ = x xy + 2 u(1 − 2uv) 1 inverse φ = u and ψ = uv − 1 . From the expression in the corollary, any symmetry generator of the Riccati equation has form Example 5.5

α

 x(xy + 1)  (xy + 2)2  1 x(xy + 1)    , −x2 ∂x +(x2 y 2 +4xy+2)∂y +β ∂y . x xy + 2 xy + 2 x2

The generators of the group g12 in noncanonical variables are obtained from this expression upon substitution of the proper values for α and β with the following result. ∂u : −x2 ∂x + (x2 y 2 + 4xy + 2)∂y , 2 )∂ , u2 ∂ : −∂ + y 2 + 4y + 2
∂ , u∂u : −x∂x + (xy 2 + 4y + x u x y x x y
2 2 1 2 ∂v : y + x ∂y , v∂v : x (xy + 1)(xy + 1)∂y , v ∂v : (xy + 1)2 ∂y . Lie gave another characterization of the most general symmetry generator of a first order ode; it is described next. Theorem 5.6 (Lie 1891) Any symmetry generator of y 0 + r(x, y) = 0 may be written in the form ρ(x, y)A + Ω(ω(x, y))U where A = ∂x − r(x, y)∂y , Aω = 0, ρ(x, y) and Ω(ω) are undetermined functions of its arguments. U is a special symmetry generator. The proof of this result may be found in Lie [113], Kapitel 7. Both Lie’s representation given in this theorem and the expression in Corollary 5.1 contain an undetermined function depending on two arguments, i.e., α or ρ respectively, and one undetermined function depending on a single argument, i.e., β and Ω. Furthermore, in order to obtain a generator explicitly, either the transformation to canonical form must be known, or the general solution of the equation at issue. Example 5.6 Coming back to the Riccati equation of Example 5.5, with 4 y + 2 , ω(x, y) = x(xy + 1) and U = ξ(x, y)∂ + η(x, y)∂ a r(x, y) = y 2 + x x y xy + 2 x2 given symmetry generator for the Riccati equation at issue, the representation for an arbitrary generator is ρ(x, y)

h

∂x −



y2 +

i  x(xy + 1)    2  4y + 2 ∂y +Ω ξ(x, y)∂x + η(x, y)∂y . x x xy + 2

This result should be compared with the representation given in the preceding Example 5.5

208 Classification of Invariant Equations. Due to the fact that invariance transformations for first order equations in general cannot be obtained algorithmically, it is of some interest to consider the inverse problem, i.e., to start from any generator and to determine the corresponding invariant equation. Such a tabulation may be applied to raise the intuition in order to guess a generator for a given first order ode. The subsequent discussion follows Lie [113], Kapitel 8; see also Cohen [32], §18 and §19 and the listing given in Table 2 on pages 231-235, and Emanuel [42], Appendix 5. The type of calculation required to obtain these tabulations is similar to the computation of invariants in the preceding section. However, now the starting points are certain simple symmetry generators for which there is a good chance of occurrence in a significant number of problems as previous experience has shown, in particular those on which the usual heuristic methods of solving first order equations are based. Its first order invariants determine the type of equation taken into account. A few examples of symmetry generators and corresponding invariant equations are given next. U = ∂x : y 0 = F (y), U = ∂y : y 0 = F (x), y y , U = x∂x − y∂y : y 0 = F (xy), U = x∂x + y∂y : y 0 = F x x y y  U = x∂x + ky∂y : y 0 = F k , x x h  y i y 1 k+ F U = φ(x)(x∂x + ky∂y ) : y 0 = . x φ(x) xk The latter result is obtained from the prolongation   U (1) = xφ∂x + kyφ∂y + kyφ0 + (kφ − φ − xφ0 )y 0 ∂y0 y xy 0 − ky φ. For F linear or quadratic in y it k and x xk comprises certain classes of Riccati or Abel equations. They will be discussed in more detail in Chapter 7 on solution algorithms. Abel’s Equation. In this subsection the symmetry properties of Abel’s equation are considered. It turns out that case i) and case ii) of Theorem 4.14 have a significantly different behavior that is described first. with the two invariants

Theorem 5.7 The two rational normal forms of Theorem 4.14 are considered separately. i) Equation y 0 +Ay 3 +By = 0 with A 6= 0 allows the two symmetry generators Z A 1 ¯ ¯ B U1 = B (∂ − By∂ ), U = y 2 ¯ dx(∂x − By∂y ) − 2 y∂y (5.18) A x A B R ¯ = exp (2 Bdx) and the commutator [U1 , U2 ] = U1 . with B

Symmetries of Differential Equations

209

ii) Equation y 0 + Ay 3 + By + 1 = 0 with A 6= 0 allows the symmetry generator  1 A0 (5.19) y∂y 3A  3 1 3B − A0 if and only if its absolute invariant K = A is a constant, A i.e., if there holds dK(A, B) = 0. (5.20) dx U=

1



A1/3

∂x −

Proof In general a first simplification is obtained by the following observation. Substituting an infinitesimal generator with coefficients ξ = ξ(x),

η = φ1 (x) + φ2 (x)y

into (5.17) and equating the coefficients of y in the resulting polynomial with zero, the coefficient of y 2 yields the equation φ1 A = 0, i.e., in general φ1 = 0. Let the simplified form of η be η = φ(x)y. For the determining system of these symmetries the lexicographic term ordering with φ > ξ is always applied, i.e., any term involving φ is considered to be higher than any term involving ξ. For case i) substitution of these values for ξ and η into (5.17) leads to 1

φ + ξ0 + 2

A0 ξ = 0, 2 A 1

(φ + Bξ)0 = 0

(5.21)

where 0 = d . Reduction with respect to the first equation yields dx h  A0  i0 0 1 1A ξ0+ − 2B ξ = 0, φ + ξ 0 + ξ = 0. A 2 2 A The former equation for ξ has the general solution Z Z Z i
h
1 ξ = exp 2 Bdx C1 A exp − 2 Bdx + C2 . A The coefficients of the generators (5.18) are obtained by choosing C1 = 0, C2 = 1 or C1 = 1, C2 = 0 respectively, applying the latter equation for R ¯ = exp (2 Bdx). determining φ and defining B For case ii) there is the additional equation φ − ξ 0 = 0. It is used for eliminating φ from system (5.21) with the result ξ0 +

A0 ξ = 0, 3 A 1

(ξ 0 + Bξ)0 = 0,

φ − ξ 0 = 0.

(5.22)

Two further reductions with respect to the first equation yield h

B−

0 0 A0  i0 1A 1A ξ = 0, ξ 0 + ξ = 0, φ + ξ = 0. 3 A 3 A 3 A 1

(5.23)

210 C and φ = 1 C A0 , i.e., the generator 3 A1/3 A A1/3 (5.19), if and only if (5.20) is satisfied. This condition is obtained from the first equation (5.22) upon substituting ξ. The latter two equations yield ξ =

The basic achievement of this theorem is the fact that certain symmetry generators with Liouvillian coefficients may be determined algorithmically. Condition (5.20) of this theorem is convenient in applications for deciding the existence of such a symmetry. It is interesting to note that it allows an explicit parametrization of all Abel equations with rational normal form of case ii) with a symmetry (5.19) in terms of an unconstrained coefficent A. To this end it may be explicitly solved for B with the result B=

1 A0 + const. · A1/3 . 3A

If condition (5.20) is not satisfied the above theorem does not apply. It is possible that different types of symmetries may exist. Example 5.7

Equation 1.46 of Kamke’s collection y 0 − x2 y 3 + 3y 2 −

1 2x − 1 y+ =0 x2 x4

(5.24)

assumes the rational normal form z 0 − x2 z 3 + 22 z = 0 by the transformation x y = z + 12 , i.e., a Bernoulli equation belonging to case i). x Example 5.8 Equation 1.41 of Kamke’s collection is y 0 + axy 3 + by 2 = 0 with constants a and b, and has rational normal form y0 +

(9a + 2b2 )2 b2 3 6a + b2 y − y + 1 = 0, 729x3 3ax

(5.25)

(3a + b2 )3 i.e., it belongs to case ii). Its invariant K = − 729 entails that 2 b (9a + 2b2 )2 there is a symmetry generator   6a + b2 ∂ + 3y ∂ 81x U= . x y ax x (9a + 2b2 )2/3 b2/3

5.3

Symmetries of Second Order Equations

In this section the objects of primary interest are not the various groups any more, but the differential equations and the symmetries attached to them. The link between these two concepts is provided by the tabulation of differential invariants given in the Section 5.1. Although it contains the complete

Symmetries of Differential Equations

211

information on all possible symmetries of any ode, the information given there is somehow organized by the wrong key for the applications that are the subject of this book. Here the symmetry problem occurs in the following form: for any given ode, determine its symmetries, and if there are any, utilize them for finding its closed form solutions. Classification of Symmetries. As a first partial answer to the symmetry problem all possible symmetry types of a quasilinear ode of order two will be determined. This is an important constraint that will turn out to be of considerable help later on for identifying the symmetry class to which a given ode belongs. An equation which is a function of the invariants of a certain group is guaranteed to allow a group of invariance transformations that comprises this group; it is possible, however, that its symmetry group is actually larger. It turns out that this does occur in a few cases. Taking these observations into account, the subsequent theorem describes the possible point symmetry groups for quasilinear ode’s of order two. The explicit form of the ode’s allowing the various symmetry groups which also follows from the listing in Section 5.1 is the subject of Chapter 6. Theorem 5.8 (Lie 1883). Any symmetry algebra of a second order quasilinear ode is similar to one in canonical variables u and v ≡ v(u) as given in the following listing. In addition the corresponding Janet basis for the determining system is given where α(u, v) and β(u, v) are the coefficients of ∂u and ∂v respectively. One-parameter group S12 : g27 = {∂v }. Janet basis {α, βu , βv }. Two-parameter groups 2 S2,1 : g26 = {∂u , ∂v }. Janet basis {αu , αv , βu , βv }. 2 S2,2 : g25 = {∂v , u∂u + v∂v }. Janet basis {αv , βu , βv − v1 β, αuu }. Three-parameter groups 2 S3,1 : g10 = {∂u + ∂v , u∂u + v∂v , u2 ∂u + v 2 ∂v }. Janet basis 2 (β − α), α − 2 α − 2 {αv , βu , βv + αu + u − uu v u − v u (u − v)2 (β − α)}. 2 S3,2 : g13 = {∂u , 2u∂u + v∂v , u2 ∂u + uv∂v }. Janet basis {αu − v2 β, αv , βv − v1 β, βuu }. 2 S3,3 (c) : g7 (c) = {∂u , ∂v , u∂u + cv∂v }, c 6= 1. Janet basis αv , βu , βv − cαu , αuu }. 2 S3,4 : g20 (r = 1) = {∂u , ∂v , u∂u + (u + v)∂v }. Janet basis {αv , βu − αu , βv − αu , αuu }. Eight-parameter group: S82 : g3 = {∂u , ∂v , u∂v , v∂v , u∂u , v∂u , u2 ∂u + uv∂v , uv∂u + v 2 ∂v }. Janet basis {αvv , βuu , βuv , βvv , αuuu , βuuv }. This listing shows in particular that there does not exist any second order ode allowing a group of point symmetries with 4, 5, 6 or 7 parameters.

212 Proof At first invariant differential equations originating from the invariants listed in Section 5.1 will be considered. For the groups gk with k = 1, 2, 3, 6, 8, 9, 11, 12, 14 and 24 an invariant of order two obviously does not exist, a second order ode with these symmetry groups is therefore excluded. For the groups with k = 16, 18, 21, 22 or 23 the parameter r is constrained to r ≥ 2 as may be seen from the listing of Section 3.4. Therefore an invariant of order two is excluded for these groups as well. The same is true for k = 19 and c 6= r. For the groups g4 and g5 the only invariant of order not higher than two has y 00 the form 0 , and there is no invariant of order one or zero, i.e., depending on y y or y 0 . As a consequence, any quasilinear equation that may be constructed from it must be linear. By Lemma 4.7 its symmetry group is the eightparameter projective group g3 . A similar argument applies to the group g15 for r = 2 with the lowest invariant of nonvanishing order D ≡ (φ1 φ02 − φ01 φ2 )y 00 − (φ1 φ03 − φ01 φ3 )y 0 + (φ01 φ002 − φ001 φ2 )y and the group g17 where D is a linear and homogeneous in y, y 0 , . . . , y (r) with constant coefficients. For the remaining values k = 7, 10, 13, 20 with r = 1, 25, 26 and 27, equations may be constructed from the given invariants with the respective symmetry groups. The only lower equation of order two is of the form y 00 = 0 with the eightparameter projective group g3 as symmetry group. This is also true for the remaining case of the group g19 with to c = r. Case 7: σ(x, y), ρ(x, y)

 PP  @ PP  PP  @  PP   @ PP )  q P R @

Case 3: σ(x), ρ(x, y)

Case 4: σ(y), ρ(x, y)

@ @ R @

@ @

Case 5: σ(x, y), ρ(x)

Case 6: σ(x, y), ρ(y)



Case 2: σ(y), ρ(x)

@ @

@ R @



Case 1: σ(x), ρ(y)

FIGURE 5.1: Arrows indicate a possible specializations. The notation Case k → Case j means that Case j is obtained by specialization from Case k. Determining the Symmetry Class. In Theorem 5.8 the symmetry groups of a second order ode are characterized in terms of canonical variables. In applications, however, they occur in actual variables. In order to apply the

Symmetries of Differential Equations

213

results of this theorem, the relation between these two descriptions has to be found. In particular this applies to the Janet bases for the determining system because it can be determined algorithmically from the given ode. This is based on the following observation. Take the Janet basis for any group defined by Theorem 5.8 in canonical variables u and v, and transform it into actual variables by means of (5.2). In general this change of variables will destroy the Janet basis property. In order to reestablish it, the algorithm JanetBasis has to be applied. During this process it may occur that a leading coefficient of an equation that is applied for reduction vanishes due to a special choice of the transformation (5.2). This has the consequence that alternatives may occur leading to different types of Janet bases. In order to obtain the complete answer, each of these alternatives has to be investigated separately. Finally all Janet bases have to be combined such that a minimal number of generic cases is retained, and all those that may be obtained from them by specialization are discarded. It turns out that up to a few exceptional cases all alternatives are due to vanishing first order partial derivatives of the transformation functions σ and ρ. Taking into account the constraint ∆ ≡ σx ρy − σy ρx 6= 0, there are altogether seven possible combinations. There exists a partial order between them described by the diagram shown in Figure 5.1. If there is only a single generic case, it must obviously be Case 7, all others may be obtained from it. Due to the size of the expressions involved, the details of the necessary calculations can only be given for the simplest groups. In order to identify a particular group type of an ode from its Janet basis, the results of Chapter 3 relating various group properties to its determining system are applied. In general this proceeds in three steps.  The first distinguishing property is the number of group parameters that is obvious from the Janet basis type.  The second distinguishing property is the number of imprimitivity systems, it is purely geometric.  Finally the algebraic structure of the corresponding Lie algebra expressed in terms of invariant properties of the coefficients of its characteristic polynomial as described in Theorem 3.7 is applied. It is part of the problem to assure that a set of criteria is obtained that will lead to a unique answer for the symmetry type of the ode at hand. For second order quasilinear ode’s the complete answer may be described as follows. Theorem 5.9 (Schwarz 1996) The following criteria provide a decision procedure for the symmetry type of a second order ode if its Janet basis in a grlex term ordering with η > ξ, y > x is given. One-parameter group (2,2)

S12 : Group g27 , Janet basis type J1,1

(2,2)

or J1,2 .

214 Two-parameter groups (2,2)

2 S2,1 : Group g26 , Lie algebra l2,2 , Janet basis type J2,3 . (2,2)

2 S2,2 : Group g27 , Lie algebra l2,1 , Janet basis type J2,3 .

Three-parameter groups 2 S3,1 : Group g10 with two systems of imprimitivity, Lie algebra l3,1 , Janet (2,2)

basis type J3,6

(2,2)

or J3,7 .

2 S3,2 : Group g13 with one system of imprimitivity, Lie algebra l3,1 , Janet basis (2,2)

(2,2)

(2,2)

type J3,4 , J3,6

or J3,7 .

2 S3,3 (c) : Group g7 (c) with two systems of imprimitivity, Lie algebra l3,2 (c), Janet (2,2)

basis type J3,6

(2,2)

or J3,7 .

2 S3,4 : Group g20 (r = 1) with one system of imprimitivity, Lie algebra l3,3 , (2,2)

Janet basis type J3,6

(2,2)

or J3,7 .

Eight-parameter group S82 : Group g3 , Janet basis type {ξyy , ηxx , ηxy , ηyy , ξxxx , ξxxy }. Proof One-parameter group. A one-parameter symmetry is uniquely determined by the respective Janet basis type. Two-parameter groups. The two groups are identified by their Lie algebra via Lemma 3.23. Three-parameter groups. At first the groups are distinguished by the number of imprimitivity systems they allow by Theorem 3.21. The two pairs corresponding to one or two systems of imprimitivity are separated w.r.t. their Lie algebras by Theorem 3.24, 3.25 and 3.26. The projective group. The eight-parameter symmetry is uniquly identified by its Janet basis type. Example 5.9

Equation 6.90 of Kamke’s collection 4x2 y 00 − x4 y 02 + 4y = 0

(5.26)

has the Janet basis η+

2y 1 ξ = 0, ξx − ξ = 0, ξy = 0 x x

(5.27)

for its point symmetries. By Theorem 5.9 it follows that it has a one-parameter symmetry group.

Symmetries of Differential Equations Example 5.10

215

Equation 6.98 of Kamke’s collection x4 y 00 − x2 y 02 − x3 y 0 + 4y 2 = 0 (2,2)

generates the type J1,2

η−

(5.28)

Janet basis

1 2y ξ = 0, ξx − ξ = 0, ξy = 0. x x

(5.29)

Consequently, it has a one-parameter symmetry. The equation 2x(x − 1) 02 4(x − 1) 0 4x2 + 5x + 2 2(x − 1) 2 2x + 1 − y=0 y − yy + y + 2 x+1 x+1 x(x + 1) x(x + 1) x(x + 1)2 (5.30) (2,2) 1 η = 0, η = 0} and generates the type J1,1 Janet basis {ξ = 0, ηx + x y therefore also a one-parameter symmetry. The close relation between the two equations (5.28) and (5.30) will become clear later on. y 00 −

Example 5.11

Equation 6.227 of Kamke’s collection (x − y 0 )y 00 + 4y 02 = 0 (2,2)

generates the type J2,3 ξx −

Janet basis

1 ξ = 0, x

ξy = 0,

ηx = 0,

1 ηy − η = 0. y

By Theorem 3.23 its Lie algebra is l2,2 , therefore by Theorem 5.9 its symmetry 2 class is S2,1 . Example 5.12

Equation 6.159 of Kamke’s collection 4y 00 y − 3y 02 − 12y 3 = 0

has the structure (5.34) with φ1 = − 15 4 , φ2 = 0, i.e., the projective group is excluded as symmetry group. The Janet basis ξx +

1 1 η = 0, ξy = 0, ηx = 0, ηy − η = 0 2y y

(2,2)

is of type J2,3 . By Theorem 3.23 its Lie algebra is l2,1 , therefore by Theo2 rem 5.9 its symmetry class is S2,2 . Example 5.13

The equation

x2 (x2 y − 2)2 y 002 − 4x4 (x2 y − 2)y 00 y 02 − 8x(x2 y − 2)y 00 y 0 +4x6 y 04 + 24x3 y 03 + 24x2 y 02 y + 16y 02 + 24xy 0 y 2 + 8y 3 = 0

216 has been written in rational form, therefore it is not quasilinear. Its Janet basis 4y 2 2y x2 y + 2 η− 2 ξ = 0, ξy = 0, ηx + x ξx − 2 x(x y − 2) x y−2 2 2x2 η − 3x y + 2 ξ = 0, x y−2 x(x2 y − 2) 2y(x2 y + 2) 2xy ξxx − 2 ξx + 2 4x 2 η + ξ=0 x y−2 (x y − 2) (x2 y − 2)2

ηy + ξx −

2

(2,2)

is of type J3,6 . By Theorem 3.21 it allows two systems of imprimitivity. By Theorem 3.25 its Lie algebra type is l3,1 . Consequently by Theorem 5.9 its 2 symmetry class is S3,1 . Example 5.14

Equation 6.133 of Kamke’s collection y 00 y + xy 00 + y 02 − y 0 = 0

generates the Janet basis 1 η − 1 ξ = 0, η − ξ + 1 η + 1 ξ = 0, ξy + ξx − x + x x y x+y x+y x+y 2 η − 2 ξ = 0, ξ − 3 ξ + 3 3 ηy + ξx − x + xx y x+y x + y x (x + y)2 η + (x + y)2 ξ = 0 (2,2)

of type J3,6 . By Theorem 3.21 it allows a single system of imprimitivity. By Theorem 3.25 its Lie algebra type is l3,1 . Consequently by Theorem 5.9 2 its symmetry class is S3,2 . Example 5.15

The equation y 00 y 0 yx6 − 2y 03 x6 + 2y 02 yx5 + y 5 = 0 (2,2)

generates the type J3,6

(5.31)

Janet basis

3 2 3 2 2 ξy = 0, ηx = 0, ηy − ξx − η + ξ = 0, ξxx − ξx + 2 ξ = 0. 2 y x x x

(5.32)

By Theorem 3.25 its Lie algebra type is l3,2 ( 32 ); therefore by Theorem 5.9 its 2 symmetry class is S3,3 ( 32 ). The equation y 0 y 00 + 2y 00 − y 04 − 12y 03 − 54y 02 − 108y 0 − 81 = 0 generates the Janet basis ξx = 0, ηx + 6ξy = 0, ηy + 5ξy = 0, ξyy = 0 (2,2)

of type J3,7 . By a similar reasoning as in the preceding case the symmetry 2 class is again S3,3 ( 23 ).

Symmetries of Differential Equations

217

2 Example 5.16 Any equation of symmetry class S3,4 must contain a term 0 proportional to an exponential in y . An example is  1 + 2y 0 y  1 1 (5.33) = 2x2 exp − y 00 y + y 02 − y 0 y − x 2x 2x (2,2)

with type J3,6

Janet basis

x 1 1 2x + 1 1 1 ξy = 0, ηx − ξx − ξ = 0, ηy − ξx + η − ξ = 0, ξxx + ξx − 2 ξ = 0. y 2xy y x x x By Theorem 3.25 its Lie algebra type is l3,3 , therefore by Theorem 5.9 its 2 symmetry class is S3,3 ( 23 ). Symmetry Type and Invariants. Tresse’s invariants and their application to equivalence problems have been discussed in Section 4.3. In the theorem below it will be shown how they may be applied for determining the symmetry class of any second order quasilinear ode. Theorem 5.10 (Tresse 1896) The following criteria provide a decision procedure for the type of symmetry group of a second order quasilinear ode in terms of its invariants under point transformations. i) If there are three functionally independent absolute invariants the equation does not have a nontrivial symmetry. ii) If there are two functionally independent absolute invariants the equation belongs to symmetry class S12 . iii) If there is a single functionally independent absolute invariant there is a two-parameter symmetry group. If for any two invariants ψ1 and ψ2 with weights r1 , s1 and r2 , s2 , respectively, there holds ∆x ψ1 ∆z ψ1 ∆y ψ1 (3r1 + 2s1 )ψ1 =0 2 + ∆x ψ2 ∆z ψ2 ∆y ψ2 (3r2 + 2s2 )ψ2 2 2 otherwise. the symmetry class is S2,1 , and S2,2

iv) If ω 4 H 6= 0 and all absolute invariants are constant there is a three2 . If for parameter symmetry group. If ω 4 = 0 the symmetry class is S3,2 any invariant ψ with weight r and s there holds ∆z ψ∆x ψ = ψ∆y ψ

32 41 (3r

+ 2s)

2 the symmetry class is S3,4 . If there holds

∆x ψ = 0, ∆z ψ = 0,

∆y ψ √ 1 6= 81 (3r + 2s) ψ 4 ω4 H

2 2 the symmetry class is S3,1 , otherwise the symmetry class is S3,3 .

218 v) If ω 4 = 0 and H = 0 the symmetry class is S82 . The invariants used in this theorem have been defined in Theorem 4.24 on page 189. The proof may again be found in Tresse’s thesis [181], page 83. In the following two examples the criteria of this theorem are applied for identifying the symmetry class of an ode. For comparison the same answer is obtained by the Janet basis analysis of Theorem 5.9 Consider the second order ode

Example 5.17

4

2

8xy 00 y 6 − 9x5 y 0 − 16xy 0 y 5 + 16y 0 y 6 = 0. Its relative invariants up to order five are 8

ω4 =

7

2187 y 0 x12 2187 y 0 x12 27x4 , H = K = , , 64 y 18 8 y 18 y6 11

H10 =

10

98415 y 0 x16 255879 y 0 x16 , H01 = . 24 64 y 128 y 24

All absolute invariants are constant, therefore the symmetry group has three parameters. Due to ω 4 = 6 0, ∆z H∆x H 80y 0 x4 + 64y 5 256 = 6= 02 4 5 H∆y H 13y x + 8y 41 and

98415 02 4 4 5 09 12 12 (y x + 5 y )y x y 6= 0 64 2 by Theorem 5.10, iv) the symmetry class S3,3 is identified. On the other hand the Janet basis for the determining system is ∆x H =

2 2 4 2 2 ξy = 0, ηx = 0, ηy − ξx − η + ξ = 0, ξxx − ξx + 2 ξ = 0. 3 y 3x x x Applying Theorem 5.9 yields the same answer. Example 5.18 As a second example for applying Tresse’s invariants consider the equation y 00 y 02 + y 00 y 2 + y 3 = 0 with the absolute invariants 8

6

4

2

8

7 0 4 23 0 6 9 8 8 0 0 2 (y 0 − 21 4 y y + 4y y − 4 y y + 4y ) y ˆ = 67108864 K , 87890625 (y 0 4 − 2y 0 2 y 2 + 1 y 4 )10 y 8 (y 0 2 + y 0 y + y 2 )6 (y 0 2 − y 0 y + y 2 )6 5 8

6

4

2

8

7 0 4 23 0 6 9 8 8 0 0 2 (y 0 − 21 4 y y + 4y y − 4 y y + 4y ) y ˆ 10 = 67108864 H . 225 (y 0 4 − 2y 0 2 y 2 + 1 y 4 )10 y 8 (y 0 2 + y 0 y + y 2 )6 (y 0 2 − y 0 y + y 2 )6 5 The remaining invariants cannot be given here due to its size. As a consequence it is difficult to apply Theorem 5.10 for determining the symmetry

Symmetries of Differential Equations

219

class of this equation. The Janet basis {ξx = 0, ξy = 0, ηx = 0, ηy − y1 η = 0} 2 immediately yields the symmetry class S2,1 .

If Tresse’s invariants are computed for a large collection of ode’s, the following general pattern appears. The calculations become extremely complex for small symmetry groups with less than three parameters. That means, despite the theoretical interest in his results, for any practical purpose it is advantageous to determine the symmetry class from the Janet basis by Theorem 5.9. This is even more true because the subsequent step in the solution algorithm, i.e., determining the canonical form, requires the knowledge of the Janet basis as well. 4 Symmetries of Lie’s Equation. The case ∂ ω 4 = 0 has been excluded in ∂y 0 Theorem 4.24. Quasilinear equations which are third order polynomials in y 0 , are called Lie’s equation. They deserve special attention due to its importance in various application fields and when the linearizability of second order ode’s is discussed. Two of its symmetry classes may be identified by the conditions given below. Theorem 5.11

Let a second order ode of the form y 00 + Ay 03 + By 02 + Cy 0 + D = 0

(5.34)

be given where A ≡ A(x, y), . . . , D ≡ D(x, y). If it allows a nontrivial point symmetry the following alternatives may occur. i) Its symmetry class is S82 iff the two relations L1 = 0, L2 = 0 are satisfied where L1 ≡ Dyy + BDy − ADx + (By − 2Ax )D + 31 Bxx − 23 Cxy + 13 C(Bx − 2Cy ), L2 ≡ 2ADy + Ay D + 13 Cyy − 32 Bxy + Axx − 13 BCy + 32 BBx − Ax C − ACx , 2 ii) Its symmetry class is S3,2 if L1 6= 0 or L2 6= 0 and Φ = Ψ = 0 where

Φ ≡ L2 (L1 L2,x − L1,x L2 ) − L1 (L1 L2,y − L1,y L2 ) −AL31 + BL21 L2 − CL1 L22 + DL32 , Ψ ≡ L2 (L1 L2,yy − L1,yy L2 ) − (2L2,y + 3AL1 − BL2 )(L1 L2,y − L1,y L2 ) −A2 L31 + (AB + Ay )L21 L2 − (AC − Ax + By )L1 L22 +(AD − 13 Bx + 23 Cy )L32 .

220 Proof The general expression for the transformation of a second order derivative v 00 (u) by the transformation u = σ(x, y), v = ρ(x, y) is v 00 =

1 {(σx ρy − σy ρx )y 00 + (σy ρyy − σyy ρy )y 03 (σx + σy y 0 )3 +[σx ρyy − σyy ρx + 2(σy ρxy − σxy ρy )]y 02 +[σy ρxx − σxx ρy + 2(σx ρxy − σxy ρx )]y + σx ρxx − σxx ρx }.

It has the structure of Lie’s equation. On the other hand, any equation with the structure (5.34) leads to a determining system of the form ξyy − 2Aηy − Bξy + Aξx − Ay η − Ax ξ = 0, ηxx − Dηy + Cηx + 2Dξx + Dy η + Dx ξ = 0, ηxy − 12 ξxx + Bηx + 32 Dξy + 21 Cξx + 12 Cy η + 12 Cx ξ = 0,

(5.35)

ηyy − 2ξxy + Bηy + 3Aηx + 2Cξy + By η + Bx ξ = 0. In order to transform it into a Janet basis, the algorithm JanetBasis described in Chapter 1 has to be applied. In the first step it yields the integrability conditions ξxxx − 3Dξxy − (2Bx − Cy − 6AD)ηx −(Dx − CD)ξy − (2Cx − 4Dy − 4BD + C 2 )ξx +(2By D − Cxy − Cy C + 2Dyy + 2Dy B)η

(5.36)

+(2Bx D − Cxx − Cx C + 2Dxy + 2Dx B)ξ = 0, ξxxy − Cξxy − ( 23 Bx − 31 Cy )ηy − (2Ax − 2AC)ηx −(Cx − Dy − BD)ξy − ( 23 Bx − 31 Cy − 3AD)ξx +(Ay D − 23 Bxy + 31 Cyy + 2Dy A)η

(5.37)

+(Ax D − 32 Bxx + 13 Cxy + 2Dx A)ξ = 0. If they are added to the original system (5.35) as the two highest equations, it has the same leading terms as the Janet basis for the projective symmetry group. In order to become a Janet basis for the group S82 , the two integrability conditions 2L2 ηy + L1 ξy + L2 ξx + L2,y η + L2,x ξ = 0, L1 ηy + L2 ηx + 2L1 ξx + L1,y η + L1,x ξ = 0 for this enlarged system are obtained. It is easy to see that they are a consequence of L1 = L2 = 0. This proves part i). The calculations for part ii) are similar, although they are more lengthy due to the larger coefficients. The first part of this theorem has been known already to Lie [109]. He gave a rather implicit criterion for S82 symmetry in terms of a set of pde’s from which

Symmetries of Differential Equations

221

the conditions L1 = L2 = 0 may be obtained as integrability conditions, see eqs. (3) and (4) in Lie [109], part II. Liouville [118] gave these constraints for the first time exlicitly in terms of the invariants L1 and L2 . Tresse [181], Theorem VII on page 56, rederived and extended this result. Since then they have been rediscovered by various authors, Grissom [63] e. g. obtained them as conditions for linearizability of a second order ode. However, the above method of proving this theorem is more transparent and in addition it provides an explicit form of the Janet basis for any equation with S82 symmetry. This remarkable result is formulated as follows. Corollary 5.2 For any second order equation (5.34) with S82 symmetry the Janet basis in grlex term ordering with η > ξ and y > x is given by the union of the systems (5.35) and (5.36), (5.37). Example 5.19 Equation 6.98 of Kamke’s collection has been considered previously in Example 5.10. It has the structure (5.34). From its coefficients the functions 4 8 2x2 − 3y , φ2 = − 5 φ1 = 6 3 x 3x are obtained. Therefore the projective group is excluded. Because Φ=−

400 5x4 + 2x2 + x , 9 (x2 + 1)10

Ψ=−

128 x2 2 81 (x + 1)11

2 the symmetry class S3,2 is excluded as well.

Example 5.20 Kamke’s equation 6.133 has been considered in Exam3 ple 6.11. By Theorem 5.11 with φ1,2 = ± , the symmetry group of (x + y)3 this equation cannot be the projective group. However both Φ = 0 and Ψ = 0 2 and the above theorem yields the symmetry class S3,2 . Example 5.21

Equation Kamke 6.180 is

y 00 (y − 1)x2 − 2y 02 x2 − 2y 0 (y − 1)x − 2y(y − 1)2 = 0.

(5.38)

2 , C = − 2 and D = − 2y(y − 1) satisfy Its coefficients A = 0, B = − y − x 1 x2 the constraints φ1 = φ2 = 0 of Theorem 5.11, i.e., its symmetry group is the projective group. A special case of Lie’s equation is obtained for A = B = C = 0, sometimes it is called generalized Emden-Fowler equation. The subsequent theorem describes its partition into symmetry classes. Theorem 5.12 The symmetry classes of the generalized Emden-Fowler equation y 00 + r(x, y) = 0 may be described as follows. i) The symmetry class is S82 if ryy = 0.

222 2 ii) The symmetry class is S3,2 if ryyy 6= 0 and
log ryyy − 65 log ryy y = 0, log rxyy −

6 5

log ryy




2 2 2 rxxyy ryyy ryy − 56 rxyy ryyy − ryyy ryy ry − 5ryyy ryy ry +

iii)

x

= 0,

25 4 3 ryy

= 0.

2 The symmetry class is S2,2 if ryyy = 0 and 3 rxxxxyy ryy −

32 2 5 rxxxyy rxyy ryy

−

43 2 2 10 rxxyy ryy

4 3 − ry − 5rxxy ryy +5rxxyy ryy

1782 4 125 rxyy

+

594 2 25 rxxyy rxyy ryy

2 2 ry ryy − 10rxyy

5 3 4 2 r − 52 ryy + 5ryy +10rxyy rxy ryy ry = 0.

iv)

The symmetry class is S12 if r obeys a differential polynomial of order eight.

Proof

The determining system for y 00 + r(x, y) = 0 is ξyy = 0, ηxx − rηy + 2rξx + ry η + rx ξ = 0, ηxy − 12 ξxx + 32 rξy = 0, ηyy − 2ξxy = 0

(5.39)

with the integrability conditions ξxxx − 3rξxy − rx ξy + 4ry ξx + 2ryy η + 2rxy ξ = 0, ξxxy + ry ξy = 0, ryy ξy = 0, 2ryy ηy + 4ryy ξx + 2ryyy η + 2rxyy ξ = 0. The first two conditions, combined with (5.39), form a Janet basis for the symmetry class S82 if and only if ryy = 0 such that the last two conditions are satisfied. This covers case i). If ryy 6= 0, these latter conditions have to be added to the system. After two autoreduction steps it contains an equation r with the leading term ryyy ξ and a further branching occurs. If ryyy 6= 0 yy x (2,2)

a type J3,6 Janet basis is generated if the conditions ii) are satisfied. By 2 Theorem 5.9 it determines uniquely the symmetry class S3,2 . If ryyy = 0, a (2,2)

type J2,3 is obtained under the constraints of case iii). By the criterion 2 given in Theorem 5.9 the symmetry class S2,2 is identified. Finally it may 2 belong to symmetry class S1 if r obeys a differential polynomial of order eight that is too complicated to be given here. Lie Algebra of Symmetry Generators. The methods for solving differential equations given by Lie [113] require the explicit knowledge of the symmetry generators. By means of the algorithms of Chapter 2, all Liouvillian generators may be obtained from the Janet basis for its coefficients. Because the symmetry type is also known from the Janet basis, the only additional step is to generate a set of generators satisfying canonical commutation

Symmetries of Differential Equations

223

relations as it is described in Chapter 3 on page 116. These steps will be explained now by some examples. Example 5.22 The two equations 6.90 and 6.98 considered above have the generators x∂x − 2y∂y and x∂x + 2y∂y respectively. This shows the close connection between the two equations although this is not obvious from their appearance. Example 5.23 Equation 6.228 has the Abelian two-parameter symmetry algebra with generators U1 = x∂x + y∂y and U2 = (x + y)∂y . Example 5.12 with equation 6.159 has the two parameter algebra U1 = ∂x , U2 = x∂x − 2y∂y with commutator [U1 , U2 ] = U1 which is already in canonical form. Example 5.24 Kamke’s equation 6.133 of Example 5.14 in symmetry class 2 S3,2 has generators U1 = ∂x − ∂y , U2 = x∂x + y∂y ,

U3 = x2 − 23 xy − 31 y 2 ∂x + 31 x2 + 23 xy − y 2 ∂y . In order to satisfy the commutation relations in canonical form, the linear ¯1 − 3 U1 , U ¯3 = U3 have to be formed. ¯2 = 2U2 and U combinations U 4 2 Example 5.25 The first equation in symmetry class S3,3 considered in Example 5.15 has symmetry generators

U1 = x∂x + 32 y∂y , U2 = y 2 ∂y , U3 = x2 ∂x . In order to obtain the canonical commutation relations of the l3,2 Lie algebra ¯1 = U2 , U ¯2 = U3 and U ¯3 = − 2 U1 have to be the linear combinations U 3 formed. The second equation of this example in the same symmetry class has generators U1 = ∂x , U2 = ∂y and U3 = y∂y − (6x + 5y)∂y . A simple algebraic ¯2 = − 1 U1 +U2 and U ¯3 = U1 +U2 − 1 U3 . ¯1 = − 1 U1 +U2 , U computation yields U 3 2 3 ¯1 , U ¯2 [= 0, [U ¯1 , U ¯3 ] = U1 and [U ¯2 , U ¯3 ] = 2 U2 . In either case there holds [U 3 Example 5.26 Equation 6.180 has been considered above in Example 5.21. Its Janet basis in grlex term ordering is 2 ξ = 0, ξyy + y − 1 y ηxx +

2 2 2y 2 − 2y 2 η − 4y − 4y ξ − 4y − 2 η + 4y − 4y ξ = 0, ηy − x x x 2 2 2 x x x x3 2 2 η − 3y − 3y ξ − 1 ξ + 1 ξ = 0, ξ − ηxy − 1 xx x y x x x2 2 y−1 x2 2 η − 4ξ + 2 ηyy − 2ξxy − y − 1 y x y y 2 − 2y + 1 η = 0,

ξxxx +

6y 2 − 6y ξxy = 0, x2

2ξ = 0 ξxxy + x xy

224 in accordance with Corollary 5.2. The special structure of this system allows it to be solved by an ad hoc method. It has a rational fundamental system leading to the vector fields U1 = x∂x , U2 = y(y − 1)∂y , U3 = x2 ∂x − x(y 2 − 1)∂y , U4 = x2 ∂x − 2xy(y − 1)∂y , U5 = x2 (y − 1)2 ∂y , U6 =

y−2 y2 2y y ∂x − 2 ∂y , U7 = y − 1 ∂x + x ∂y , x(y − 1) x y 2y 2 U8 = y − 1 ∂x − x ∂y .

5.4

Symmetries of Nonlinear Third Order Equations

The methods described in the preceding section may be extended to higher order equations more or less straightforwardly, although the larger number of symmetry types leads to more alternatives that have to be considered. For various reasons it appears to be appropriate to distinguish equations that are equivalent to a linear one from those that are not. These latter equations are simply called genuinely nonlinear third order equations; they are the subject of the current section whereas the linearizable equations are considered in Section 5.5. Classification of Symmetries. The main result of this section, the equivalent of Theorem 5.8 for nonlinear third order equations, is given first. Theorem 5.13 Any symmetry algebra of a genuinely nonlinear third order quasilinear ode is similar to one in canonical variables u and v ≡ v(u) as given in the following listing. In addition the corresponding Janet basis is given where α(u, v) and β(u, v) are the coefficients of ∂u and ∂v respectively. One-parameter group S13 : g27 = {∂v }. Janet basis {α, βu , βv }. Two-parameter groups 3 S2,1 : g26 = {∂u , ∂v }. Janet basis {αu , αv , βu , βv }. 3 S2,2 : g25 = {u∂u + v∂v , ∂v }. Janet basis {αv , βu , βv − v1 β, αuu }. 3 S2,3 : g4 = {v∂v , ∂v }. Janet basis {α, βu , βvv }. 3 S2,4 : g15 (r = 2) = {∂v , u∂v }. Janet basis {α, βv , βuu }. Three-parameter groups 3 S3,1 : g10 = {∂u + ∂v , u∂u + v∂v , u2 ∂u + v 2 ∂v }. Janet basis 2 (β − α), α − 2 α − 2 {αv , βu , βv + αu + u − uu v u − v u (u − v)2 (β − α)}. 3 S3,2 : g13 = {∂u , 2u∂u + v∂v , u2 ∂u + uv∂v }. Janet basis {αu − v2 β, αv , βv − v1 β, βuu }.

Symmetries of Differential Equations

225

3 S3,3 (c) : g7 = {∂u , ∂v , u∂u + cv∂v }, c 6= 0, 1. Janet basis αv , βu , βv − cαu , αuu }. 3 S3,4 : g20 (r = 1) = {∂u , ∂v , u∂u + (u + v)∂v }. Janet basis {αv , βu − αu , βv − αu , αuu }. 3 S3,5 : g5 = {∂u , ∂v , v∂v }. Janet basis {αu , αv , βu , βvv }. 3 S3,6 : g17 (l = 1, ρ1 = 1, α1 = 0) = {∂u , ∂v , u∂v }. Janet basis {αu , αv , βv , βuu }. 3 S3,7 : g17 (l = 1, ρ1 = 1, α1 = 1) = {eu ∂v , ueu ∂v , ∂u }. Janet basis {αu , αv , βv , βuu − 2βu + β}. 3 S3,8 : g17 (l = 2, ρ1 = ρ2 = 0, α1 = 0, α2 = 1) = {eu ∂v , ∂v , ∂u }. Janet basis {αu , αv , βv , βuu − βu }. 3 S3,9 (c) : g17 (l = 2, ρ1 = ρ2 = 0, α1 = 1, α2 = c) = {eu ∂v , ecu ∂v , ∂u }, c 6= 0, 1. Janet basis {αu , αv , βv , βuu − (a + 1)βu + aβ}. 3 S3,10 : g24 = {∂u , ∂v , u∂u + v∂v }. Janet basis {αv , βu , βv − αu , αuu }. Four-parameter groups 3 S4,1 : g6 = {∂u , ∂v , v∂v , u∂u }. Janet basis {αv , βu , αuu , βvv }. 3 S4,2 : g14 = {∂u , u∂u , v∂v , u2 ∂u + uv∂v }. Janet basis {αv , βv − v1 β, αuu − v2 βu , βuu }. 3 S4,3 (c) : g19 (r = 2) = {∂u , ∂v , u∂v , u∂u + cv∂v }, c 6= 2 Janet basis {αv , βv − cαu , αuu , βuu }. 3 S4,4 : g20 (r = 2) = {∂u , ∂v , u∂v , u∂u + (u2 + 2v)∂v }. Janet basis {αv , βv − 2αu , αuu , βuu − 2αu }. Groups with six parameters S63 : g12 = {∂u , ∂v , u∂u , v∂v , u2 ∂u , v 2 ∂v }. Janet basis {αv , βu , αuuu , βvvv }.

Proof At first invariant differential equations originating from the invariants listed in Section 5.1 will be considered. For the groups gk with k = 1, 2, 3, 11 and 12 an invariant of order three obviously does not exist. A third order ode with these symmetry groups is therefore excluded. For the groups with k = 21, 22 or 23 the parameter r is constrained to r ≥ 2 as it may be seen from the listing of Section 3.4. Therefore, an invariant of order three is excluded for these groups as well. The quasilinear equation y 000 y 0 − 23 y 002 + r(x) = 0 corresponding to g8 is invariant under the larger group g12 ⊃ g8 . Therefore its symmetry group is g12 . A similar argument applies to group g9 . (15) The group g15 for r = 2 allows the three lowest invariants Φ1 = x, φ1 φ01 φ001 φ1 φ01 φ000 1 (15) (15) 0 00 0 000 Φ2 = D ≡ φ2 φ2 φ2 and Φ3 = D1 ≡ φ2 φ2 φ2 y y 0 y 00 y y 0 y 000 as it may be seen from the listing in Section 5.1. The special choice φ1 = 1 (15) (15) and φ2 = x yields the simplification Φ2 = y 00 and Φ3 = y 000 from which a (15) nonlinear third order equation may be constructed. For r = 3, again Φ1 = x and φ1 φ01 φ001 φ000 1 (15) 0 00 000 Φ2 = φ2 φ2 φ2 φ2 . y y 0 y 00 y 000

226 (15)

Choosing φ1 and φ2 as above and leaving φ3 undetermined yields now Φ2 = φ00 y 000 − φ000 y 00 . It is not possible to construct a nonlinear third order equation from it which is linear in y 000 and is therefore excluded; see, however, Theorem 5.16. (16) (16) 1 The group g16 for r = 2 has the lowest invariants Φ1 = x and Φ2 = D D (16)

with D and D1 as above. Applying the same simplification leads to Φ2 = y 000 . Again it is not possible to construct a nonlinear third order equation y 00 from it which is linear in y 000 and is therefore excluded. For the remaining values k = 4, . . . , 7, 9, 10, 12, 13, 14, 19, 20 and 24, . . . , 27 equations may be constructed from the given invariants with the respective symmetry groups as listed above. The groups g17 have at least three parameters. There exist exactly four of them for this minimal value; they have been listed in Section 3.4. Setting up the system (5.14) for these groups, the two lowest invariants of the form 3 3 Φ, Φ0 are obtained. For the groups S3,6 , . . . , S3,9 one gets Φ = v 00 , Φ = 00 0 00 0 00 0 v − 2v + v, Φ = v − v and Φ = v − (α + 1)v + αv respectively. In each case, a nonlinear third order equation that is linear in v 000 may be constructed. Proceeding in a similar way for the seven types of four-parameter groups g17 listed in Section 3.4 yields lowest invariants of the form v 000 + c1 v 00 + c2 v 0 + c3 v with constants ci from which a nonlinear equation, linear in v 000 , cannot be constructed. Therefore these groups are excluded. The lower equations for k = 1, . . . , 10, 13, 14, 20, 23, . . . , 27 are excluded because its order is different from three. For k = 19, 21 and 22 the lower equation is linear, they are covered by Theorem 5.16 below. The lower equations for g11 ⊂ g12 are already subsumed under the equation with g12 symmetry as listed above. In his book on differential equations Lie [113] gives in Chapter 22, §3 a complete listing of three parameter groups of a two-dimensional manifold. It is instructive to identify the symmetry groups of the above theorem in 3 Lie’s list. For symmetry classes S3,k , k = 1, 2, . . . , 6 this is fairly obvious, the corresponding groups are 1), 2), 4), 8), 10) and 12) in his enumeration 3 respectively. The symmetry type of class S3,7 is similar to Lie’s group 9), the transformations functions are σ = −x, ρ = ye−x ; the symmetry type of class 3 S3,8 is similar to Lie’s group 11), the transformation functions are σ = − log x, y 3 ρ = x . Finally the symmetry type of S3,9 is similar to Lie’s group 5) with 1 transformation functions σ = c − 1 log x, ρ = yx1(c−1) and α = c. In the subsequent Section 5.5 it will be shown that equations with a fiveor seven-parameter symmetry group are always linearizable, therefore they do not occur in the above theorem. Determining the Symmetry Class. The preceding theorem describes the possible symmetry types of third order quasilinear ode’s in canonical coordinates. Based on this result, the next theorem allows it to determine the

Symmetries of Differential Equations

227

symmetry type of any such equation in actual variables from the Janet basis of its determining system. In addition to the Janet bases that occurred for second order ode’s in the previous section, various Janet bases of order four are required. They are given in Appendix B on page 380. Theorem 5.14 The following criteria provide a decision procedure for determining the symmetry class of a third order ode if its Janet basis in a grlex term ordering with η > ξ, y > x is given. One-parameter group (2,2)

S13 : Janet basis of type J1,1

(2,2)

or J1,2 .

Two-parameter groups 3 S2,1 : Group g26 with a one-parameter system of imprimitivity, Lie algebra (2,2)

l2,2 , Janet basis type J2,3 . 3 S2,2 : Group g27 with a one-parameter system of imprimitivity, Lie algebra (2,2)

l2,1 , Janet basis type J2,3 . 3 S2,3 : Group g4 with two systems of imprimitivity, Lie algebra l2,1 , Janet basis (2,2)

(2,2)

type J2,1 , J2,2

(2,2)

or J2,4 .

3 S2,4 : Group g15 (r = 2) with one system of imprimitivity, Lie algebra l2,1 , (2,2)

Janet basis type J2,1

(2,2)

or J2,5 .

Three-parameter groups 3 S3,1 : Group g10 with two systems of imprimitivity, Lie algebra l3,1 , Janet (2,2)

basis type J3,6

(2,2)

or J3,7 .

3 S3,2 : Group g13 with one system of imprimitivity, Lie algebra l3,1 , Janet basis (2,2)

(2,2)

type J3,4 , J3,6

(2,2)

or J3,7 .

3 S3,3 (c) : Group g7 (c) with two systems of imprimitivity, Lie algebra l3,2 (c), (2,2)

Janet basis type J3,6

(2,2)

or J3,7 .

3 S3,4 : Group g20 (r = 1) with one system of imprimitivity, Lie algebra l3,3 , (2,2)

Janet basis type J3,6

(2,2)

with c1 6= 1 or J3,7

with c1 6= 0.

3 : Group g5 with two systems of imprimitivity, Lie algebra l3,4 , Janet basis S3,5 (2,2)

(2,2)

(2,2)

type J3,4 , J3,5 , J3,6

(2,2)

or J3,7 .

3 S3,6 : Group g17 (l = 1, ρ1 = 1, α1 = 0) with one system of imprimitivity, Lie (2,2)

(2,2)

algebra l3,5 , Janet basis type J3,4 , J3,6

(2,2)

or J3,7 .

228 3 S3,7 : Group g17 (l = 1, ρ1 = 1, α1 = 1) with one system of imprimitivity, Lie (2,2)

(2,2)

algebra l3,3 , Janet basis type J3,4 , J3,6 c1 = 0.

(2,2)

with c1 = 1 or J3,7

with

3 S3,8 : Group g17 (l = 2, ρ1 = ρ2 = 0, α1 = 0, α2 = 1) with one system of (2,2)

(2,2)

imprimitivity, Lie algebra l3,4 , Janet basis type J3,4 , J3,6

(2,2)

or J3,7 .

3 S3,9 (c) : Group g17 (l = 2, ρ1 = ρ2 = 0, α1 = 1, α2 = c) with one system of (2,2)

(2,2)

imprimitivity, Lie algebra l3,2 (c), Janet basis type J3,4 , J3,6

(2,2)

or J3,7 .

3 S3,10 : Group g24 with a one-parameter system of imprimitivity, Lie algebra (2,2)

l3,2 (c = 1), Janet basis type J3,6 . Four-parameter groups 3 S4,1 : Group g6 with two systems of imprimitivity, Lie algebra l4,10 , Janet (2,2)

(2,2)

(2,2)

basis type J4,9 , J4,10 or J4,14 . 3 S4,2 : Group g14 with one system of imprimitivity, Lie algebra l4,1 , Janet basis (2,2)

type J4,9

(2,2)

or J4,12 .

3 S4,3 (c) : Group g19 (r = 2) with one system of imprimitivity, Lie algebra (2,2)

l4,5 (c), Janet basis type J4,9

(2,2)

or J4,14 .

3 S4,4 : Group g20 (r = 2) with one system of imprimitivity, Lie algebra l4,4 , (2,2)

Janet basis type J4,9

(2,2)

or J4,14 .

Six-parameter group S63 : Janet basis of type {ξy , ηx , ξxxx , ηyyy }, type {ξy , ηy , ξxxx , ηxxx }, type {ηx , ηy , ξxy , ξxxx , ξyyy } or type {ηx , ηy , ξyy , ξxxx , ξxxy }. Proof One-parameter group. A one-parameter symmetry is uniquely determined by the Janet basis type. Two-parameter groups. The first two symmetry classes are singled out by their one-parameter system of imprimitivity by Theorem 3.20. The further distinction between the two groups is obtained through their Lie algebra by (2,2) Theorem 3.23. For Janet basis type J2,1 the symmetry class is uniquely identified by Theorem 3.20 through their systems of imprimitivity. If the (2,2) (2,2) (2,2) Janet basis type is J2,2 , J2,4 or J2,5 the symmetry class is unique. (2,2)

Three-parameter groups. If there is a type J3,5 3 S3,5

Janet basis, the symmetry (2,2)

class is uniquely identified. For Janet basis type J3,4 , by Theorem 3.21 3 again symmetry class S3,5 is uniquely identified by its two systems of imprimitivity. The remaining cases allow a single system of imprimitivity; the

Symmetries of Differential Equations

229

symmetry type is identified from its Lie algebra by means of Theorem 3.24. (2,2) 3 For Janet basis type J3,6 , Theorem 3.21 gives the unique answer S3,10 if a one-parameter system of imprimitivity is found; otherwise it combines the 3 symmetry classes into those allowing two systems of imprimitivity, i.e., S3,1 , 3 3 S3,3 (c) and S3,5 , and the remaining ones. In the former case Theorem 3.25 leads to a unique identification by its Lie algebra. In the latter the same 3 3 is true except for the two symmetry classes S3,4 and S3,7 , both with a l3,3 Lie algebra. They are distinguished by their different connectivity properties which are identified by c1 6= 1 or c1 = 1 respectively. For Janet basis type (2,2) J3,7 the discussion is identical; Theorems 3.21 and 3.26 are applied now. 3 3 The distinction between symmetry classes S3,4 and S3,7 is obtained by the condition c1 6= 0 or c1 = 0 respectively. 3 3 Four-parameter groups. The symmetry classes S4,1 and S4,2 are uniquely (2,2)

(2,2)

identified from the Janet basis types J4,10 and J4,12 . For Janet basis type (2,2)

3 J4,9 , by Theorem 3.22, symmetry class S4,1 is uniquely identified by its two systems of imprimitivity. The remaining three symmetry types are distinguished by their Lie algebra applying Theorem 3.27. For Janet basis type (2,2) 3 J4,14 , by Theorem 3.22, again symmetry class S4,1 is uniquely identified through its two systems of imprimitivity. The distinction between the two remaining cases is obtained from Theorem 3.28. Six-parameter group. The only symmetry class S63 comprising six parameters is identified from any of the Janet bases of order six without constraints on its coefficients.

For later use some algebraic properties of the various symmetry algebras are given next. Corollary 5.3 All one- and two-dimensional symmetry algebras, all 3 three-dimensional symmetry algebras except those of symmetry classes S3,1 3 and S3,2 that are simple and of type l3,1 , and all four-dimensional symmetry 3 algebras except that of symmetry class S4,1 of type l4,1 are solvable. Proof The solvable cases follow immediately from the Definition 3.6 3 of solvability. The nonsolvable algebra of S4,1 has the Levi decomposition ¯ with radical R generated by {y∂y } and the simple Levi factor of type R⊕S l3,1 generated by {∂x , x∂x + 12 y∂y , x2 ∂x + xy∂y }. The six-dimensional algebra g12 is semi-simple with the direct-sum decomposition {∂u , u∂u , u2 ∂u } ⊕ {∂v , v∂v , v 2 ∂v } into two simple type l3,1 algebras. Example 5.27

Equation 7.2 in Kamke’s collection x2 y 000 + xy 00 (y − 1) + xy 02 − y 0 (y − 1) = 0 (2,2)

generates the type J1,2 to symmetry class S13 .

 1 ξ, ξ ; consequently it belongs Janet basis η, ξx − x y

230 Example 5.28

The equation

y 000 y 0 y − 3y 002 y + y 00 y 0 (2xyy 0 + 3y 0 + 2y 2 ) − y 03 (y 02 + 2xy 0 + 4y) = 0  Janet basis ξx + y1 η, ξy + y1 ξ, ηx , ηy with nonvanishing coefficients a1 = b2 = y1 . By Theorem 3.23 it generates a l2,2 Lie 3 algebra, therefore by Theorem 5.14 its symmetry class is S2,1 . (2,2)

generates the type J2,3

Example 5.29 Equation 7.7 from Kamke’s collection yy 000 −y 0 y 00 +y 2 y 0 = 0  (2,2) generates the type J2,3 Janet basis ξx + y1 η, ξy , ηx , ηy − y1 η with nonvanishing coefficients a1 = y1 , d1 = − y1 . By Theorem 3.23 it generates a l2,1 Lie 3 algebra; consequently, by Theorem 5.14 its symmetry class is S2,2 . Example 5.30

The equation

y 000 y 03 x3 + x3 y 003 − 3x(2y 0 + x)(xy 0 − 2)y 0 y 00 − x3 (y + 1)y 06 − 6xy 04 + 8y 03 = 0 (2,2)

generates the type J2,4 3 class S2,3 . Example 5.31 y 000 y 02 −(3y 0 +

Janet basis; consequently it belongs to symmetry

The equation

2 04 6 03 4y 02  4 002 0  02 6 0 4y 00 2 06 )y +y y +x y + y − y + y − 2y + 4y =0 x x x2 x3 x x (2,2)

generates the type J2,5 3 class S2,4 . Example 5.32

Janet basis; consequently it belongs to symmetry

The equation

(y 000 y 0 − 4y 002 )(y − x)2 + 10y 00 y 0 (y 0 + 1)(y − x) − y 02 (10y 02 + 3y 0 + 10) = 0 (2,2)

generates a type J3,6

Janet basis for its symmetry generators with nonvan2 , d = − 2 and d = −d = ishing coefficients c1 = 1, c2 = −c3 = x − 3 2 y 1 x−y 2 . By Theorem 3.21 it allows two systems of imprimitivity. By Theo(x − y)2 rem 3.25, its Lie algebra is l3,1 . Consequently, by Theorem 5.14 its symmetry 3 . class is S3,1 Example 5.33

The equation

 2 3y 2  00 0 4 1 05  2 6y 5 03 y 000 y 0 y 6 −3y 002 y 6 + 6y 0 y+ 2 y 0 + y y y − 5 y −2 3y+ 2 y 04 y 3 − y =0 x x x x x (2,2)

generates a type J3,7

Janet basis for its symmetry generators with nonvan1 , c = − 2 , c = − 2 and d = 2 . By Theorem 3.22 ishing coefficients a3 = − x 2 1 y 3 x y

Symmetries of Differential Equations

231

it allows a single system of imprimitivity, by Theorem 3.26 its Lie algebra is 3 l3,1 . Consequently, by Theorem 5.14 its symmetry class is S3,2 . Example 5.34 y 000 +

The equation 1
1 3 00
03 3 02 3 y y − y y + 2 y0 y2 − 3 y3 + 8 = 0 x x x x x (2,2)

generates a type J3,6 Janet basis for its symmetry generators with coeffiy cients a1 = 0, b1 = 2x , c1 = − 32 . By Theorem 3.21 it allows two systems of imprimitivity. By Theorem 3.25 its Lie algebra is l3,2 ( 32 ). Consequently, by 3 Theorem 5.14 its symmetry class is S3,3 ( 23 ). 0

Example 5.35 The equation y 000 y 02 − 3y 002 y 0 − y 002 − y 06 e−2/y = 0 gen(2,2) erates a type J3,6 Janet basis for its symmetry generators with coefficients a1 = c1 = −1. By Theorem 3.21 it allows a single system of imprimitivity. By Theorem 3.25 its Lie algebra is l3,3 . Consequently, by Theorem 5.14 its 3 symmetry class is S3,4 . Example 5.36

The equation 2

2

x2 y 000 y 0 y 4 − 4x2 y 00 y 4 + 10x2 y 00 y 0 y 3 − 10xy 00 y 0 y 4 4

4

3

2

−10x2 y 0 y 2 + 2x2 y 0 + 20xy 0 y 3 − 10y 0 y 4 = 0 (2,2)

generates a type J3,6

Janet basis for its symmetry generators with nonva2 and d = 2 . By Theorem 3.21 it nishing coefficients c2 = − y2 , d1 = − x 3 x2 allows two systems of imprimitivity. By Theorem 3.25 its Lie algebra is l3,4 . 3 Consequently, by Theorem 5.14 its symmetry class is S3,5 . Example 5.37 Consider the following four equations generating type (2,2) J3,4 Janet bases for its symmetry generators, the nonvanishing coefficients of which are also given. x10 y 000 y 0 + 2x9 y 000 y 0 + x8 (6x − 1)y 002 + 2x7 (9x − 2)y 00 y 0 + 4x6 (3x − 1)y 02 − 1 = 0 2 and d = 2 . with nonvanishing coefficients a3 = − x 1 x x10 y 000 y 00 + 2x8 (x + 1)y 000 y 0 + x6 y 000 y + x8 (6x + 1)y 00

2

+x6 (18x2 + 16x + 1)y 00 y 0 + 6x5 y 00 y 2

+2x4 (x + 1)(6x2 + 2x − 1)y 0 + 3x2 (2x2 − 1)y 0 y − y 2 − 1 = 0 2 , d = 2 + 2 and d = 1 . with nonvanishing coefficients a3 = − x 1 2 x x2 x4 2

x10 y 000 y 00 + x8 (2x + 1)y 000 y 0 + 6x9 y 00 + x6 (18x2 + 6x − 1)y 00 y 0 2

+x4 (6x2 − 1)(2x + 1)y 0 − 1 = 0

232 2 and d = 2 + 1 . with nonvanishing coefficients a3 = − x 1 x x2 x10 y 000 y 00 + 2x8 (x + 2)y 000 y 0 + 3x6 y 000 y + 3x8 (2x + 1)y 00

2

+x6 (18x2 + 36x + 11)y 00 y 0 + 6x4 (3x + 1)y 00 y 2

+2x4 (x + 2)(6x2 + 6x − 1)y 0 + 3x2 (6x2 + 4x − 5)y 0 y − 9y 2 − 1 = 0 2 , d = 2 + 4 and d = 2 . There with nonvanishing coefficients a3 = − x 1 2 x x2 x4 (2,2) are altogether six possible symmetry classes for the Janet basis type J3,4 . By Theorem 3.21 it follows that in all four cases there is a single system 3 of imprimitivity; consequently, symmetry class S3,5 is excluded. By Theo1 rem 3.24 the Lie algebras l3,5 , l3,3 , l3,4 and l3,2 ( 2 ) are identified. From the 3 3 3 classification in Theorem 5.14 the symmetry classes are S3,6 , S3,7 , S3,8 and 3 S3,9 respectively. Example 5.38

Consider the equation

x2 y 000 y 0 + x6 y 002 y 02 + x2 y 002 + 4x5 y 00 y 03 + 10xy 00 y 0 + 4x4 y 04 + 10y 02 = 0. (2,2)

It generates a type J3,6

Janet basis for its symmetry generators with nonva2 , d = − 2 and d = 2 . By Theorem 3.21 nishing coefficients c1 = −1, c3 = x 1 3 x x2 it allows a one-parameter system of imprimitivity, by Theorem 5.14 its sym3 metry class S3,10 is identified. Example 5.39

Consider the equation

1 y
− y 002 y 3 3y 0 + 1 y
y 000 y 0 y 3 y 0 + x 2x
04 2 −y 00 y 02 y 2 y 0 + 11 y + 4xy 06 + 16y 05 y + 22 x x y y = 0. (2,2)

It generates a type J4,14 Janet basis for its symmetry generators with nony vanishing coefficients a1 = a2 = a3 = a4 = 0, b1 = x and b2 = −1. By Theorem 3.22 it allows two systems of imprimitivity, therefore by Theorem 5.14 3 its symmetry class is S4,1 . Example 5.40

Consider the equation

3
y 00 y 000 − x8 y 003 − 6x6 (2xy 0 + y)y 002 − 3 16x6 y 02 + 16x5 y 0 y + 4x4 y 2 − x
−64x5 y 03 − 96x4 y 02 y − 6 8x3 y 2 − 32 y 0 − 8x2 y 3 + 63 y = 0 x x (2,2)

with Janet basis type J4,9 for its symmetry generators with nonvanishing coefficients b2 = − 23 , b4 , c2 , c4 and all d0i s. By Theorem 3.22 it allows a single system of imprimitivity, by Theorem 3.28 its Lie algebra is l4,5 ( 32 ). 3 Consequently, by Theorem 5.14 its symmetry class is S4,3 ( 32 ).

Symmetries of Differential Equations

233

Equations with a Six-Parameter Symmetry. Although equations with this symmetry type are covered by the above theorem, a different proceeding will be described next. It does not require the Janet basis for the determining system. Later on, for third order equations allowing a seven-parameter symmetry group, a similar method will be applied exclusively. Theorem 5.15 If a quasilinear third order ode belongs to symmetry class S63 , either of the following two cases applies where Aj ≡ Aj (x, y) and Bk ≡ Bk (x, y) for all j and k. i) The equation has the structure (A0 y 02 + A1 y 0 + 1)y 000 − (3A0 y 0 + 23 A1 )y 002 + (A2 y 03 + A3 y 02 + A4 y 0 + A5 )y 00 +B1 y 06 + B2 y 05 + B3 y 04 + B4 y 03 + B5 y 02 + B6 y 0 + B7 = 0. (5.40) Necessary coefficient constraints for symmetry class S63 are as follows. A0 A1,x − 21 A0 A1 A5 + 13 A0 A4 − 14 A1,x A21 +

1 3 12 A1 A5

− 16 A1 A3 + 31 A2 = 0,

A20 A5 + 3A0 A1,y − 12 A0 A21 A5 − 12 A0 A1 A4 + A0 A3 − 34 A1,y A21 + 14 A31 A4 − 12 A21 A3 + 12 A1 A2 = 0, A20 A1 A5,x − 2A20 A5,y − 32 A0 A1,y A1 A5 + A0 A1,y A4 − 14 A0 A31 A5,x + 41 A0 A31 A25 − 12 A0 A21 A4,x − 14 A0 A21 A4 A5 + A0 A21 A5,y + A0 A1 A3,x − 12 A0 A1 A3 A5 + 16 A0 A1 A24 − 2A0 A2,x + A0 A2 A5 + 38 A1,y A31 A5 − 14 A1,y A21 A4 + 18 A41 A4,x − 18 A41 A4 A5 − 81 A41 A5,y − 14 A31 A3,x + + 12 A21 A2,x − 34 A21 A2 A5 − 31 A21 A3 A4 + A30 A5,x −

13 2 2 4 A0 A1 A5,x

− 21 A20 A1 A4,x +

−A20 A4,y + 16 A20 A24 + 32 A0 A21,y +

5 1 3 3 2 12 A1 A3 A5 + 12 A1 A4 1 1 1 2 2 A1 A2 A4 + 6 A1 A3 − 3 A2 A3 =

13 2 2 A0 A1 A5,y

33 2 8 A0 A1,y A1 A5

+ A20 A3,x − 72 A0 A1,y A1 A4

+3A0 A1,y A3 + 34 A0 A41 A5,x −

17 4 2 24 A0 A1 A5

3 − 25 8 A0 A1 A5,y −

13 2 4 A0 A1 A3,x

+ 76 A0 A21 A3 A5 + 14 A0 A21 A4,y −

+ 11 2 A0 A1 A2,x −

31 12 A0 A1 A2 A5

+ 14 A1,y A1 A2 − 38 A51 A4,x + 4 − 29 24 A1 A3 A5 −

13 4 2 48 A1 A4

−

1 2 3 + 31 24 A1 A3 A4 + 4 A1 A2,y −

+

13 3 8 A0 A1 A4,x

+

19 3 24 A0 A1 A4 A5 13 2 2 24 A0 A1 A4

+ 21 A0 A1 A3,y − 13 A0 A1 A3 A4 − A0 A2,y

17 5 48 A1 A4 A5

11 3 8 A1 A2,x

+

43 2 24 A1 A2 A4

+ 38 A51 A5,y + 34 A41 A3,x 13 3 6 A1 A2 A5

− A21 A23 +

− 18 A31 A3,y

25 12 A1 A2 A3

− 56 A22 = 0,

B5 − 52 B6 A1 − A0 B7 + 4A21 B7 + 21 A1 A5,x + 12 A1 A25 − 13 A4 A5 − A5,y = 0, B4 − 2B6 A0 − 2B6 A21 + 4A0 A1 B7 + 13 A0 A5,x + 29 A0 A25 + 4A31 B7 + 23 A21 A5,x + 94 A21 A25 + 16 A1 A4,x − 32 A1 A5,y − 13 A3 A5 − 13 A4,y − 19 A24 = 0,

0,

234 B3 − 3B6 A0 A1 − 12 B6 A31 + A20 B7 + 7A0 A21 B7 +

13 12 A0 A1 A5,x

+ 23 A0 A1 A25

+ 16 A0 A4,x − 29 A0 A4 A5 − 35 A0 A5,y + A41 B7 + 16 A31 A5,x + 19 A31 A25 + 18 A21 A4,x −

11 2 24 A1 A5,y

1 12 A1 A3,x

+

1 18 A1 A3 A5

−

− 13 A1 A4,y

1 + 18 A1 A24 − 29 A2 A5 − 16 A3,y − 29 A3 A4 = 0,

B2 A0 − 14 B2 A21 − B6 A30 − 43 B6 A20 A21 + 14 B6 A0 A41 + 4A30 A1 B7 + A20 A31 B7 + 12 A20 A1 A4,x − 21 A20 A1 A5,y − 13 A20 A3,x −

1 2 2 12 A0 A4

+ 34 A0 A21,y

3 A0 A1,y A21 A5 − 43 A0 A1,y A1 A4 + 13 A0 A1,y A3 − 12 A0 A51 B7 − 16 5 A0 A41 A25 + − 144

1 3 24 A0 A1 A4,x

− 14 A0 A21 A3,x −

1 2 72 A0 A1 A3 A5

31 3 144 A0 A1 A4 A5

+

1 2 2 48 A0 A1 A4

+

−

1 3 24 A0 A1 A5,y

+ A0 A1 A2,x

1 1 1 1 − 43 72 A0 A1 A2 A5 − 3 A0 A1 A3,y − 9 A0 A1 A3 A4 + 3 A0 A2,y − 6 A0 A2 A4 3 − 16 A21,y A21 +

1 4 32 A1,y A1 A5

1 A51 A4,x − − 24

1 5 96 A1 A4 A5

5 − 96 A41 A24 − 41 A31 A2,x + 1 A21 A2,y − − 12

7 3 32 A1,y A1 A4

+

+

1 5 24 A1 A5,y

25 3 144 A1 A2 A5

11 2 144 A1 A2 A4

−

1 2 2 72 A1 A3

1 4 12 A1 A3,x

+

+

7 2 48 A1,y A1 A3

−

1 3 12 A1 A3,y

−

11 72 A1 A2 A3

5 4 144 A1 A3 A5

−

+

+ 18 A1,y A1 A2

5 3 36 A1 A3 A4 13 2 36 A2

+

= 0,

B1 − 12 B2 A1 + 12 B6 A0 A31 + A30 B7 − A20 A21 B7 + 16 A20 A4,x − 12 A20 A5,y − 18 A0 A1,y A1 A5 +

1 12 A0 A1,y A4

13 A0 A31 A25 − 16 A0 A21 A4,x − − 144

− A0 A41 B7 −

1 2 48 A0 A1 A4 A5

5 + 72 A0 A1 A3 A5 + 16 A0 A1 A4,y +

− 16 A0 A3,y +

1 3 32 A1,y A1 A5

−

1 A41 A5,y − − 96

1 3 48 A1 A3,x

1 + 12 A21 A3,y −

1 2 18 A1 A3 A4

1 2 72 A0 A1 A4

1 2 48 A1,y A1 A4

+

+

3 3 16 A0 A1 A5,x

+

13 2 24 A0 A1 A5,y

+ 16 A0 A2,x −

1 4 96 A1 A4,x

−

1 4 96 A1 A4 A5

−

1 2 24 A1 A2,x

5 3 144 A1 A3 A5

+

1 3 2 144 A1 A4

11 72 A1 A2 A4

+

5 2 72 A1 A3

+

5 36 A0 A2 A5

−

5 36 A2 A3

= 0.

ii) The equation has the structure (A0 y 02 + y 0 )y 000 − (3A0 y 0 + 23 )y 002 + (A2 y 03 + A3 y 02 )y 00 +B1 y 06 + B2 y 05 + B3 y 04 + B4 y 03 + B5 y 02 = 0.

(5.41)

Its symmetry class is S63 if and only if its coefficients satisfy the following

Symmetries of Differential Equations

235

constraints. B1 − B2 A0 + B3 A20 − B4 A30 + B5 A40 − 16 A23 A20 + 31 A3 A2 A0 − 16 A22 = 0, A0,x + 31 A3 = 0, A0,y + 13 A2 = 0, A2,y − 2A2,x A0 − 3B2 + 6B3 A0 − 8B4 A20 + 8B5 A30 − 34 A23 A0 + 43 A3 A2 = 0, A3,x − B4 + 4B5 A0 = 0, A3,y − A2,x = 0, B4,x − 3B5,y − B5,x A0 − 23 B5 A3 = 0, B3,x − B4,y − 2B5,y A0 − 43 B5 A2 = 0, A2,x,x − B4,y + 4B5,y A0 − 43 B5 A2 = 0, B2,xx − 13 B4,yy − 83 B5,yy A0 + 49 B4,y A3 − 49 B5,y A3 A0 −

20 9 B5,y A2

8 4 2 − 10 9 A2,x B4 + 9 A2,x B5 A0 − 3 B2 B5 + 3 B3 B4

− 89 B42 A0 + 89 B4 B5 A20 −

4 2 27 B4 A3

+

8 27 B5 A3 A2

= 0.

The proof of this theorem is similar to that of Theorem 5.11. Starting from a generic equation with structure (5.40) or (5.41) that is obtained from the canonical form v 000 v 0 − 23 v 002 = 0 (see Theorem 6.16) by a general point transformation, a Janet basis involving the undetermined functions Aj and Bk is generated. The above constraints assure that this Janet basis corresponds to a six-parameter symmetry. The intermediate calculations are very lengthy and are therefore omitted. The advantage of these criteria is that they are very fast. Example 5.41

Consider the equation

2

2

3

2

2x3 y 000 y 0 y − 2xy 000 y 3 − 6x3 y 00 y 0 y + 6x3 y 00 y 0 − 6x2 y 00 y 0 y 4

3

2

+6xy 00 y 0 y 2 − 6y 00 y 3 + 6x2 y 0 − 12xy 0 y + 6y 0 y 2 = 0 with the structure (5.40). Its coefficients 2 2 3, A0 = − x2 , A1 = 0, A2 = − 3x3 , A3 = 3x2 , A4 = − y3 , A5 = x y y y 3 , B =B =0 B1 = B2 = 0, B3 = − 3x3 , B4 = 62 , B5 = − xy 6 7 y y

satisfy the necessary constraints of Theorem 5.15, case i). Furthermore, the Janet basis y2 1 ξ = 0, ξy = 0, ηy − ξx − y1 η + x x2 2 1ξ = 0 ξy,y − x2 ξx,x + y1 ξy + x2 ξx − 12 ξ = 0, ξxxx = 0, ξxxy − x x,y y y y ηx −

is generated. Consequently, it belongs to symmetry class S63 . If the above ode is slightly changed, e. g. by adding the term y, i.e., B7 = y, the symmetry

236 is destroyed. This may be discovered by testing the coefficient constraints of Theorem 5.15 which is very fast, avoiding in this way the much more elaborate Janet basis calculation. Example 5.42 Consider the equation i h h i x 3x y 0 + 3 y 002 y 02 + y 0 y 000 − 2 (x − 1)y (x − 1)y h i 3x(x − 2) 04 3x 3 03 02 + y y y + y 00 + (x − 1)y 2 (x − 1)2 y (x − 1)2 y 2 +

2(x2 − 3x + 3) 03 x2 − 4x + 6 02 y + y =0 (x − 1)2 y 2(x − 1)2

with the structure (5.41). Its coefficients 3 A0 = xy x− y , A2 23x 2 , A3 = 2 , B1 = B2 = 0, xy − y x y − 2xy + y B3 =

3x2 − 6x 2x2 − 6x + 6 , B = x2 − 4x + 6 5 2 2 , B4 = 2 x y − 2xy + y x y − 2xy + y 2x2 − 4x + 2 2 2

satisfy the constraints of Theorem 5.15, case ii). Consequently, it belongs to symmetry class S63 .

5.5

Symmetries of Linearizable Equations

For order higher than two, linearizable equations comprise more than a single equivalence class as it has been shown in Theorem 4.9. Moreover, they combine into several symmetry classes as it will be seen in this section. Because the methods for solving linear equations are different from the nonlinear ones, a special section is devoted to them. At first equations of third order are considered; the results for them are more detailed than for arbitrary order. Classification of Symmetries. At first a complete survey of all possible symmetry types is provided. Theorem 5.16 Any symmetry algebra of a third order linearizable ode is similar to one in canonical variables u and v ≡ v(u) as given in the following listing. In addition the corresponding Janet basis is given where α(u, v) and β(u, v) are the coefficients of ∂u and ∂v respectively. Four-parameter symmetry 3 S4,5 : g16 (r = 3) = {∂v , u∂v , φ(u)∂v , v∂v }.  φ000 Janet basis α, βuv , βvv , βuuu − 00 βuu . φ Five-parameter symmetry

Symmetries of Differential Equations

237

3 S5,1 : g18 (l = 2, ρ1 = 1, ρ2 = 0, α1 = 0, α2 = 1) = {∂v , u∂v , eu ∂v , v∂v , ∂u }. Janet basis: {αu , αv , βuv , βvv , βuuu − βuu }. 3 S5,2 (a) ≡ g18 (l = 3, ρ1 = ρ2 = ρ3 = 0, α1 = 0, α2 = 1, α3 = a) = {∂v , eu ∂v , eau ∂v , v∂v , ∂u }, a 6= 0, 1. Janet basis: {αu , αv , βuv , βvv , βuuu − (a + 1)βuu + aβu }. Seven-parameter symmetry S73 : g23 (r = 3) = {∂v , u∂v , u2 ∂v , v∂v , ∂u , u∂u , u2 ∂u + 2uv∂v }. Janet basis: {αv , βuv − αuu , βvv , αuuu , βuuu }.

Proof The equivalence classes of third order equations with a linear representative have been determined in Theorem 4.9. The canonical equation v 000 = 0 for E13 generates the Janet basis of the seven-parameter group g23 (r = 3) given in the listing on page 142. It corresponds to symmetry class S73 . The canonical equation v 000 + v = 0 for E23 has√been shown to be equivalent to v¯000 − (a + 1)¯ v 00 + a¯ v = 0 with a = 12 (1 ± i 2) in Lemma 4.3. √ Similarly, the canonical equations v 000 + cv 0 + v = 0 for c 6= 0, c 6= − 23 3 2 of E33 (c) have been shown to be equivalent to equations of the same form if c and a are related by (4.32). Equations of this latter form generate the Janet basis {αu , αv , βuv , βvv , βuuu − (a + 1)βuu + aβu }. From the listing on page 142 the group g18 (l = 3, ρ1 = ρ2 = ρ3 = 0, α1 = 0, α2 = 1, α3 = a) is 3 identified as a symmetry group. It corresponds to symmetry class S5,2 (a). If √ 3 3 3 000 00 c = − 2 2 in the canonical form of E3 (c), it is equivalent to v − v = 0 with Janet basis {αu , αv , βuv , βvv , βuuu − βuu }. From the listing on page 142 group g18 (l = 2, ρ1 = 1, ρ2 = 0, α1 = 0, α2 = 1) is identified as a symmetry group 3 corresponding to symmetry class S5,1 . Finally the canonical representative 000 3 v + Q(x)v = 0 of E4 (Q) generates the Janet basis {α, βuv , βvv , βuuu + Qβ}. 3 From the listing on page 142 the group g16 (r = 3) for symmetry class S4,5 is obtained. In Section 4.1 on page 172 the equivalence classes of linearizable third order equations have been completely identified. The subsequent corollary combines these results with the symmetry classes obtained in the preceding theorem. Corollary 5.4 The symmetry classes of linear third order equations are composed of equivalence classes as follows. √ 3 ≡ E33 (− 32 2), S73 ≡ E13 , S5,1 √ 3 3 S5,2 (a) ≡ E23 ∪ E33 (c) with c 6= − 23 2, S4,5 ≡ E43 (Q). The constants a and c for the five-parameter symmetries are related by (4.32). The symmetries identified above will be applied in the next chapter for solving the corresponding equations. To this end the algebraic structure of these symmetry algebras is described next.

238 Corollary 5.5 The symmetry algebras of the four- and five-dimensional symmetries are solvable. The seven-dimensional algebra has the Levi decomposition R >
S with radical R = {∂v , u∂v , u2 ∂v , v∂v } of type l4,6 (a = b = 1) and the simple Levi factor S = {∂u , u∂u + v∂v , u2 ∂u + 2uv∂v } of type l3,1 . Proof The solvable cases follow immediately from the Definition 3.6 of solvability, the seven-dimensional algebra has been considered before in Example 3.6. Determining the Symmetry Class. Based on the above classification of symmetries, the subsequent theorem provides algorithmic means for determining the symmetry class of any given quasilinear linearizable third order ode. The Janet basis types required for its identification are listed in Appendix B on page 380. Theorem 5.17 The following criteria provide a decision procedure for identifying the symmetry class of a linearizable third order ode if its Janet basis in grlex term ordering with η > ξ, y > x is given. Four-parameter symmetry (2,2)

(2,2)

(2,2)

3 S4,5 : Janet basis of type J4,2 , type J4,17 or type J4,19 . Five-parameter symmetries 3 3 S5,1 and S5,2 (a) : Three Janet basis types may occur. In all three cases the parameter a determining the symmetry type is a solution of the equation

P 2 (a2 − a + 1)3 + Q2 R(a − 21 )2 (a + 1)2 (a − 2)2 ) = 0.

(5.42)

3 3 If a 6= 0 the symmetry class is S5,2 (a) and S5,1 otherwise. Its coefficients P , Q and R for the various cases are given below. (2,2)

i) Janet basis type J5,1 . P = − 92 a1,x a1 + 13 a1,x e1 + c3,x − a31 − 23 a21 e1 − 2a1 c3 − 31 a1 e21 + a1 e3 + c1 e2 − Q = a1,x − 2a21 − a1 e1 − 3c3 − 31 e21 + e3 ,

2 3 27 e1


+ 13 e1 e3 − e4 ,

R = 3Q.

(2,2)

ii) Janet basis type J5,2 . P = − 92 b1,y b1 + 31 b1,y e1 + d2,y − b31 − 23 b21 e1 − 2b1 d2 − 13 b1 e21 + b1 e2 + c1 e3 − Q = b1,y − 2b21 − b1 e1 − 3d2 − 31 e21 + e2 ,

2 3 27 e1


+ 13 e1 e2 − e5 ,

R = 3Q.

Symmetries of Differential Equations

239

(2,2)

iii) Janet basis type J5,3 . P =

9 2

a3,x a3 c31 − 13 a3,x c21 e1 + a3,x c1 c3 + c5,x + a2 a23 c31 − 31 a2 a3 c21 e1 +a2 a3 c1 c3 − a2 c5 + 2a33 c41 − a23 c31 e1 + 3a23 c21 c3 + a3 c31 e2 + 13 a3 c21 e21 − a3 c21 e3 + 2a3 c1 c5 + a3 c4 − 31 c21 e1 e2
2 +c1 c3 e2 − 27 c1 e31 + 31 c2 e1 e3 − c1 e5 − e4 ,

Q = a3,x c21 + a2 a3 c21 + 3a23 c31 − a3 c21 e1 + 3a3 c1 c3 + c21 e2 + 13 c1 e21 − c1 e3 + 3c5 ,

R = c31 Q.

Seven-parameter symmetry S73 : Janet basis of type {ξy , ηxy , ηyy , ξxxx , ηxxx }, type {ηx , ξxx , ηyy , ξxyy , ξyyy } or type {ηy , ξyy , ηxx , ξxxx , ξxxy }. (2,2)

(2,2)

(2,2)

Proof A Janet basis of type J4,2 , J4,17 or J4,19 identifies the sym3 metry class S4,5 uniquely because these Janet basis types do not occur for any other four-parameter symmetry as it may be seen from Theorem 5.14. If (2,2) the equations between the generic coefficients of a Janet basis of type J5,1 , (2,2)

(2,2)

J5,2 or J5,3 and the transformation functions σ(x, y) and ρ(x, y) are transformed into a Janet basis, one of the equations is (5.42) from which the value 3 3 of a and thereby the symmetry class S5,2 (a) may be obtained if a 6= 0, or S5,1 if a = 0. A Janet basis allowing a seven-parameter symmetry group identifies uniquely symmetry class S73 . 3 In the first three examples equations in symmetry class S4,5 are discussed. As it will turn out in the next chapter, they are all equivalent to each other.

Example 5.43

The equation

y 000 y 2 − 6y 00 y 0 y + (2,2)

generates the type J4,2

3 00 2 8x 6 y y + 6y 03 − y 02 y − y0 y2 = 0 x x 2x − 1

Janet basis ξ = 0, ηxy − y2 ηx = 0,

ηyy − y2 ηy + 22 η = 0, y

3 η − 8x η = 0. ηxxx + x xx 2x − 1 x

3 By case a) of the above theorem it belongs to symmetry class S4,5 .

Example 5.44

The equation

(y 000 y 0 − 3y 002 )y 3 (y − 12 ) − 3y 00 y 02 y 2 (y − 21 ) −y 05 x(8y 3 − 6y + 3) + y 04 y(4y 3 − 6y + 3) = 0

240 (2,2)

generates the type J4,19 Janet basis η = 0, ξxx = 0, ξxy = 0, x(8y 3 − 6y + 3) 4y 3 − 6y + 3 8y 3 − 6y + 3 ξy − ξy + ξ = 0. ξyyy − y3 ξyy − 3 3 1 1 y (y − 2 ) y (y − 2 ) y 3 (y − 21 ) 3 By case b) of the above theorem it belongs to symmetry class S4,5 .

The equation

Example 5.45

y 000 (y 0 + 1)(y + x − 21 ) − 3y 002 (y + x − 12 ) − 4y 05 x + 4y 04 (y − 4x) +8y 03 (2y − 3x) + 8y 02 (3y − 2x) + 4y 0 (4y − x) + 4y = 0 (2,2)

generates the type J4,17 Janet basis η + ξ = 0, ξxy − ξxx = 0, ξyy − ξxx = 0, ξxxx −

4y 4x 4 ξy − ξx + ξ = 0. x + y − 12 x + y − 21 x + y − 21

3 By case c) of the above theorem it belongs to symmetry class S4,5 .

Example 5.46

The equation

1 y 000 y 2 − 3y 00 2 y − xy 00 y 0 2 y + 3y 00 y 0 2 − 2y 00 y 0 y 2 − 12 y 00 y 0 y y 000 y 0 y + x x 2 1 3 15 1 00 3 00 2 04 03 03 − x y y − 2 y y + 2xy + 3y y + x y − x y 0 y 2 x 2 − 32 y 0 y − 32 y 0 y 3 − 93 y 0 y 2 − 13 y 4 − 34 y 3 = 0 x x x x x 2,2 Janet basis generates the type J5,3

y 1 η = 0, ηx + x ξx + x

y 1 ξ = 0, ηy + x ξy + x

1 3 1 2 ξxy − x y ξxx − x ξy + y ξx + y 2 η − xy ξ = 0, 2 ξyy − x2 ξxx + y3 ξy + 3x2 ξx + 2x3 η − 12 ξ = 0, y y y y

ξxxx −

xy + 6 xy 2 + 3y 3xy + 15 2xy + 9 xy + 6 ξy + ξx + η− ξ = 0. x ξxx + x3 x2 x2 y x3

By the above theorem, a five-parameter symmetry group is assured. From the Janet basis coefficients P = − 31 xy 2 , Q = − 31 xy and R = xy are obtained. Substituting these values into (5.42) leads to a2 (a−1) = 0 with the zero a = 0 3 corresponding to the symmetry class S5,1 .

Symmetries of Differential Equations Example 5.47

241

The equation 2

2

2

5

y 000 y 0 y 3 − 3y 00 y 3 − 4y 00 y 0 y 3 − 3y 00 y 0 y 2 + 3xy 0 y 2 5

5

4

4

4

+8xy 0 y + 6xy 0 − 3y 0 y 3 − 8y 0 y 2 − 6y 0 y = 0 2,2 generates the type J5,2 Janet basis

ηx = 0, ηy = 0, ξxx = 0, ξxy + 12 η = 0, y 3y 2 + 8y + 6 3xy 2 + 8xy + 6x 4y + 3 ξy + ξx ξyyy − y ξyy + y2 y3 3xy 2 + 16xy + 18x 3y 2 + 8y + 6 + η− ξ = 0. 4 y y3 7 The Janet basis coefficients yield P = − 10 3 , Q = − 3 xy and R = −7. With these values (5.42) leads to

(a + 2)(a + 21 )(a − 13 )(a − 23 )(a − 32 )(a − 3) = 0. The proper choice for a will be determined in the subsequent chapter. Third Order Equations with Maximal Symmetry. Similar to second order equations with projective symmetry, third order equations with symmetry groups comprising seven parameters deserve special attention. Due to the fact that their canonical form v 000 = 0 does not contain parameters or undetermined functions, equations with this symmetry type have a special structure from which the S73 symmetry may be identified without generating a Janet basis. The answer has been given essentially by Zorawski [192], a student of Lie at the end of the 19th century. As usual, σ(x, y) and ρ(x, y) are the transformation functions from a canonical form to the actual variables x and y. Theorem 5.18 (Zorawski 1897, Neumer 1929) A quasilinear third order ode belongs to symmetry class S73 if and only if either of the following two cases applies where Aj ≡ Aj (x, y) and Bk ≡ Bk (x, y) for all j and k. a) If σx 6= 0 the equation has the structure (A0 y 0 + 1)y 000 − 3A0 y 002 + (A1 y 02 + A2 y 0 + A3 )y 00 +B1 y 05 + B2 y 04 + B3 y 03 + B4 y 02 + B5 y 0 + B6 = 0. The coefficients satisfy the constraints A0,xx − 61 A0,x A3 − 16 A0 A3,x + 16 A2,x − 16 A3,y = 0, A0,y − A0,x A0 + 16 A20 A3 − 16 A0 A2 + 61 A1 = 0,

(5.43)

242 B5,x + 23 B5 A3 − 2B6,y − 3B6,x A0 − 5B6 A0,x − 2B6 A0 A3 − 32 B6 A2 − 13 A3,xx − 23 A3,x A3 −

4 3 27 A3

= 0,

B5,y − 2B5 A0,x + 32 B5 A0 A3 − 7B6,y A0 + 2B6,x A20 + 5B6 A0,x A0 − 76 B6 A20 A3 − 32 B6 A0 A2 + 56 B6 A1 + 43 A0,x A3,x 1 2 + 11 18 A0,x A3 + 3 A0 A3,xx −

+ 13 A2,xx +

1 18 A2,x A3

−

1 4 3 18 A0 A3,x A3 − 27 A0 A3 4 13 3 A3,x,y − 18 A3,y A3 = 0,

B4 − 4B5 A0 + + A0 A3,x + 23 A0 A23 − 13 A2 A3 − A3,y = 0, 23 2 2 B3 − 6B5 A20 + 20B6 A30 + 37 A20 A3,x + 18 A0 A3 + 13 A0 A2,x 5 5 A0 A2 A3 − 37 A0 A3,y − 18 A1 A3 − 13 A2,y − 19 A22 = − 18 10B6 A20

0,

11 3 17 3 2 1 2 6 A0 A3,x + 18 A0 A3 + 2 A0 A2,x 5 1 1 2 − 36 A20 A2 A3 − 11 6 A0 A3,y + 6 A0 A1,x − 6 A0 A1 A3 1 7 A0 A22 − 16 A1,y − 36 A1 A2 = 0, − 12 A0 A2,y − 36

B2 − 4B5 A30 + 15B6 A40 +

B1 − B5 A40 + 4B6 A50 + 12 A40 A3,x + 14 A40 A23 + 16 A30 A2,x 1 A30 A2 A3 − 21 A30 A3,y + 16 A20 A1,x − − 36

− 16 A20 A2,y − 16 A0 A1,y −

1 36 A0 A1 A2

−

1 2 18 A0 A1 A3 1 2 12 A1 = 0.

b) If σx = 0 the equation has the structure y 0 y 000 − 3y 002 + (A1 y 02 + A2 y 0 )y 00 + B1 y 05 + B2 y 04 + B3 y 03 + B4 y 02 = 0. (5.44) The coefficients satisfy the constraints B4 + 31 A2,x + 19 A22 = 0, B3 + A1,x + 13 A1 A2 = 0, A2,y − A1,x = 0, A1,xy + 23 A1 A1,x + B2,x = 0, A1,yy + 3B2,y − 6B1,x + 2A1 A1,y + 94 A31 + 2(A1 B2 − A2 B1 ) = 0. Proof If the expressions for the derivatives up to third order (5.3), (5.5) and (5.6) (see also (B.6), (B.7)) are substituted into the canonical form v 000 = 0, and the resulting expression is separated w.r.t. derivatives of y, the structure (5.43) or (5.44) follows. For any equation obtained in this way, equating the coefficients of the monomials in the derivatives to the coefficients Aj and Bk of (5.43) or (5.44) leads to a system of equations that is obeyed by the transformation functions. The above constraints are obtained by transforming this system into a Janet basis. If they are satisfied it is guaranteed that a general solution involving seven constants does exist. In the subsequent chapter the systems of pde’s determining the transformation functions to canonical form will be studied in more detail. In particular, its relation to certain invariants will become apparent.

Symmetries of Differential Equations

243

Neumer (1929) has actually given more complete results on equations with the structure (5.43) or (5.44). On pages 20-23 he determines criteria for identifying a four-, five- or seven-dimensional symmetry in terms of its coefficients. However, in the former two cases they are considerably more complicated, and the Janet bases are required anyhow for determining the transformation functions to the equivalent linear equation. Therefore they are not given here. A special case of this type of results for identifying an enlarged symmetry group of a linear equation is discussed in the Exercise 5.8. Example 5.48

Equation 7.8 from Kamke’s collection y 000 y 2 − 29 y 00 y 0 y +

15 03 4 y

=0

9 and B = 15 . has structure (5.43) with nonvanishing coefficients A2 = − 2y 3 4y 2 They satisfy the necessary constraints of Theorem 5.18, case a); consequently, this equation belongs to symmetry class S73 . Alternatively, the symmetry class may be read off from its Janet basis ξy = 0, ηxy − ξxx −

3 ηx = 0, 2y

3 η + 3 η = 0, ξ ηyy − 2y y xxx = 0, ηxxx = 0. 2y 2 Example 5.49

Consider the equation

1 y 000 y 2 − 3y 00 2 y + 3y 00 y 0 2 − 12 y 00 y 0 y − 3 y 00 y 2 y 000 y 0 y + x x x2 3 02 9 0 2 3 3 03 + 15 x y − x2 y y − x3 y y − x4 y = 0 with the structure (5.43). Its coefficients 3x −12 −3 A0 = x y , A1 = y 2 , A2 = y , A3 = x , −3y −9 B1 = B2 = 0, B3 = 152 , B4 = −3 xy B5 = x2 , B6 = x3 y satisfy the necessary constraints of Theorem 5.18, case a); consequently it belongs to symmetry class S73 . Example 5.50

Consider the equation

2 2 2 5 4 y 000 y 0 − 3y 00 + y9 y 00 y 0 + 3x − 6x + 6 y 00 y 0 − 3 6x y 0 − 182 y 0 x(x − 1) y (x − 1) y 2 3 2 3 + 6x − 6 y 0 2 = 0 − 9x − 18x + 18 y 0 − x − 3x 2 xy(x − 1) x (x − 1)

244 with the structure (5.44). Its coefficients 2 , A1 = y9 , A2 = 3x − 6x + 6 , B1 = 3 −6x , B2 = −18 x(x − 1) y (x − 1) y2 2 3 2 B3 = −9x + 18x − 18 , B4 = −x +23x − 6x + 6 xy(x − 1) x (x − 1)

satisfy the constraints of Theorem 5.18, case b); consequently, its symmetry class is S73 . Linear Equations of Any Order. The special structure of these equations entails that their symmetries may be described in much more detail than for a general quasilinear equation. Amazingly they have been studied only recently by Krause and Michel [95, 96], and Mahomed and Leach [129]. Let a linear homogeneous equation of n-th order be given in rational normal form by L ≡ y (n) + q2 y (n−2) + q3 y (n−3) + . . . + qn−1 y 0 + qn y = 0.

(5.45)

The following theorem describes the possible symmetries of these equations. Theorem 5.19 (Mahomed and Leach 1990, Krause and Michel 1988, 1991) The symmetry algebra of a linear nth order differential equation (5.45) with fundamental system y1 , y2 , . . . , yn has one of three possible forms. i) A (n+4)-parameter symmetry algebra {y1 ∂y , y2 ∂y , . . . , yn ∂y , y∂y } >
sl2 . This case applies if and only if (5.45) is equivalent to v (n) = 0. ii) A (n + 2)-parameter symmetry algebra {∂x , y1 ∂y , y2 ∂y , . . . , yn ∂y , y∂y }. This case applies if and only if (5.45) is equivalent to a linear equation with constant coefficients. iii) A (n + 1)-parameter symmetry algebra {y1 ∂y , y2 ∂y , . . . , yn ∂y , y∂y }. It cannot have a n + 3-dimensional symmetry algebra. The proof of this theorem is based on a direct computation of the vector fields generating the symmetries. The details may be found in the articles by Mahomed [129] or Krause and Michel [96]. If an equation has maximal symmetry, the transformation to v (n) = 0 is actually achieved by the LaguerreForsyth transformation. The special case of third order equations has been treated above in Theorem 5.17. Linear equations with maximal symmetry group have a very special structure, in fact they may be parametrized by a single independent coefficient as the following result shows. Theorem 5.20 (Mahomed and Leach 1990) For linear equations (5.45) with maximal symmetry, the coefficient p of y (n−2) determines the equation

Symmetries of Differential Equations

245

completely. For n not higher than five these equations are y 000 + py 0 + 21 p0 y = 0, y (4) + py 00 + p0 y 0 +

3 00 10 p


9 2 + 100 p y = 0,

4 2 0 9 00 4 y (5) + py 000 + 32 p0 y 00 + 10 p + 25 p y + 15 p000 + 25 pp0 y = 0. In the article by Mahomed [129] these equations are given up to order eight. The severe constraints on the coefficients of an equation with maximal symmetry described in this theorem have their counterpart in a very special structure of the solutions of such equations as it is shown next. Theorem 5.21 (Krause and Michel 1988) The linear equations with maximal symmetry of Theorem 5.20 are the symmetric power of the second order s n−1 1
pz = 0, i.e., L ≡ M ; a fundamental source equation M ≡ z 00 + n + 1 3

system therefore has the form yk = z1k z2n−1−k for k = 0, . . . , n − 1 where z1 , z2 is a fundamental system for M . This result reduces the problem of solving a n-th order equation with maximal symmetry to a second order problem. Example 5.51

The equation

y (4) −

5 2

+

20  00 40 0  9 9 2 y + 3 y + 16 + 2 y = 0 x x x

(5.46)

has a maximal, i.e., an eight-parameter symmetry group. The source equation
2 ) exp (∓ x ). z 00 − 14 + 22 z = 0 has the two independent solutions z1,2 = (1± x 2 x By the above theorem a fundamental system for (5.46) is y1 =

(x − 2)3 3x (x + 2)(x − 2)2 x exp ( ), y = exp ( ), 2 2 3 2 2 x x

(x + 2)2 (x − 2) x (x + 2)3 3x exp (− ), y4 = exp (− ). 3 2 2 x x3 It is left as Exercise 5.9 to obtain the same answer from the Loewy decomposition of (5.46). y3 =

246

Exercises Exercise 5.1

Prove the relation between U (n) and D given in Lemma 5.2.

Exercise 5.2 Let y 00 + r(x)y = 0 be a second order linear ode in rational normal form with the general solution y = C1 y1 + C2 y2 . Determine its eight symmetry generators in terms of this fundamental system. Exercise 5.3 Show that any linear homogeneous ode of order n for y ≡ y(x) with fundamental system y1 , y2 , . . . , yn has n + 1 symmetry generators yk (x)∂y , k = 1, . . . , n, and y∂y . Exercise 5.4 The form of the two equations (2.67) is constrained by the requirements that the complete system for w is linear and homogeneous and that no new integrability conditions are introduced. Determine their most general form in conformity with these requirements. 3 3 Exercise 5.5 S3,1 , . . . , S3,10 of Theorem 5.13 coincide with the listing given in Lie [113], pages 501-502. Why is this list only partially included in Theorem 5.8 ?

Exercise 5.6 Determine the structure invariance group for the canonical
00 3 . form equation v 000 + v 0 r v 0 , u = 0 of the symmetry class S2,3 v Exercise 5.7

3 The same for v 000 + r(u, v 00 ) = 0 and symmetry class S2,4 .

Exercise 5.8 Show that y 000 + A(x)y 0 + B(x)y = 0 allows a five-parameter symmetry group if and only if one of the following two conditions is satisfied. 6

 Θ0 2 Θ000 Θ0 3 Θ003 Θ00 Θ0 28  Θ03 3 3 +A 3 − A0 = 0 − 7 3 − 9A = 0 or −4 3 3 + Θ3 Θ3 Θ3 Θ3 Θ3 9 Θ3 Θ3 2

where Θ3 is defined in Theorem 4.9. Exercise 5.9 decomposition.

Determine the fundamental system of (5.46) from its Loewy

Chapter 6 Transformation to Canonical Form

It is assumed now that a differential equation of order two or three allows a nontrivial symmetry of a known type or, if its order is one, at least a single symmetry generator is explicitly known. This knowledge is utilized for transforming it into a canonical form corresponding to its symmetry class whenever this is possible. To this end a system of equations is constructed the solutions of which are the desired transformation functions. These equations must be such that algorithms are available for determining its solutions in well-defined function fields, e. g. rational or Liouvillian functions. A complete understanding of these equations is achieved by determining for each symmetry class the structure invariance group of its canonical form equations, and the differential invariants that may be constructed from the transformation functions.

6.1

First Order Equations

Any first order equation y 0 + r(x, y) = 0 is equivalent to the canonical form v 0 = 0 according to Theorem 4.10. However, in general there is no algorithm available for determining the transformation functions that actually generate it. Furthermore, it is not guaranteed that a symmetry generator may be explicitly obtained despite the fact that infinitely many do exist as has been explained in Section 5.2. As a consequence, in order to obtain a manageable problem it must be assumed that at least a single symmetry generator is a priori known. That means, the first order ode together with one or more symmetry generators is the input to the problem at issue. The following theorem, which is due to Lie [113], Kapitel 6, § 5, describes how a canonical form for the given ode may be obtained if a single nontrivial symmetry generator is known. Theorem 6.1 (Lie 1881) Let the first order ode y 0 + r(x, y) = 0 allow the symmetry generator U = ξ(x, y)∂x + η(x, y)∂y . If canonical variables u and v may be determined such that it represents the translation ∂v , the ode assumes the form v 0 + s(u) = 0 in these variables.

247

248 Proof The first prolongation of the canonical generator ∂v allows the invariant Φ(u, v 0 ). From this the given form of the canonical ode follows immediately. dx = dy as ξ(x, y) η(x, y) required by Lemma 3.4. If it fails, a canonical form may not be obtained. In this case the symmetry may be applied to generate an integrating factor as will be explained in Section 7.1. The crucial step in this theorem is to solve the ode

Example 6.1

Equation 1.15 from the collection by Kamke y 0 + y 2 − 2x2 y + x4 − 2x − 1 = 0

allows the symmetry generator U = ∂x + 2x∂y . By Lemma 3.4 the transfordy = 2x mation function σ(x, y) to canonical variables is obtained by solving dx 2 with the result u ≡ σ(x, y) = y − x , then v ≡ ρ(x, y) follows from Z ρ=

√

√ dy = y−u = x. u=σ y − u u=σ

The inverse functions x = v, y = u + v 2 yield y 0 = 2v + 10 and finally lead to v the canonical form v 0 + 2 1 = 0. u −1 If two generators corresponding to the groups g26 or g25 are known, the transformation to canonical form is always possible as is shown next. Theorem 6.2 Let the first order ode y 0 + r(x, y) = 0 allow two symmetry generators Ui = ξi (x, y)∂x + ηi (x, y)∂y for i = 1, 2. If they generate the group g25 or g26 , a Liouvillian transformation according to Lemma 3.5 transforms it to v 0 + a = 0 where a is a constant. Proof The canonical form is an immediate consequence of the invariants given in the listing in Chapter 5.1 on page 205. The constant a introduced in this theorem is not an integration constant. Its value is uniquely tied up with the particular canonical form. The system of equations for the transformation functions given in Lemma 3.5, and the solution algorithms described in Lemma 2.9 and Lemma 2.10, imply that knowing two symmetry generators for a first order ode y 0 + r(x, y) = 0 guarantees that a transformation to canonical form may be achieved by Liouvillian functions which may always be determined. (x − y)2 + 1 = 0 allows the two symmetry (x − y)2 − 1 generators U1 = ∂x + ∂y and U2 = y∂x + x∂y of a type g25 group. Its transformation to canonical form has been obtained in Example 3.28 with the Example 6.2

The equation y 0 +

Transformation to Canonical Form

249

1 and y ≡ ψ(u, v) = u − v − 1 . It yields result x ≡ φ(u, v) = u − v + 2v 2v 2 2 0 2v + (1 − 2v )v 2 0 0 and finally v = 3 . y = 2 2v − (1 + 2v 2 )v 0

6.2

Second Order Equations

Let a second order ode be given in actual variables x and y ≡ y(x). In general it will be assumed that it is polynomial in the derivatives, and rational in the dependent and the independent variables. In addition the dependence on y 00 is assumed to be linear if it is not stated otherwise explicitly. For any equation of this kind with a nontrivial symmetry new variables u and v(u) are searched for such that the equation is transformed to a canonical form corresponding to its symmetry class. This process consists of two steps. i) A system of equations is constructed that is obeyed by the desired transformation functions. ii) Solutions of these equations in well-defined function fields have to be determined. A priori there is no guarantee that the desired transformation functions may be found. However, due to the special structure of these equations, by the methods that have been described in Chapter 2 it is assured that large classes of transformation functions may be obtained algorithmically. A complete listing of possible equation types and the corresponding classes of solutions that may be obtained for them is given next, it is a partial summary of the results of this section. In order to arrive at these results, Lemma 2.9, 2.10 and 2.11 are applied expressing a solution in terms of integrals. Theorem 6.3 In order to transform a second order quasilinear ode with a nontrivial symmetry to canonical form, the type of equations and the function fields for which solution algorithms are available may be described as follows. S12 : First order ode, in general no algorithm available. 2 S2,1 : Third order system

zxx +a1 zy +a2 zx = 0, zxy +b1 zy +b2 zx = 0, zyy +c1 zy +c2 zy = 0. (6.1) Hyperexponential solutions may be determined. 2 S2,2 : Liouvillian functions explicitly known. 2 S3,1 : Partial Riccati-like system

zx + a1 z 2 + a2 z + a3 = 0, zy + b1 z 2 + b2 z + b3 = 0. Liouvillian functions in terms of integrals over rational solutions for z.

250 2 S3,2 : Partial Riccati-like system

zxx + a1 zx2 + a2 zx + a3 = 0, zy + b1 zx + b2 = 0. Liouvillian functions in terms of integrals over rational solutions for z. 2 2 S3,3 and S3,4 : Liouvillian functions over the base field explicitly known.

S82 : Third order system (6.1) with coefficients that are rational solutions of partial Riccati-like systems z1,x + z12 + a1 z1 + a2 z2 + a3 = 0, z1,y + z1 z2 + b1 z1 + b2 z2 + b3 = 0, z2,x + z1 z2 + c1 z1 + c2 z2 + c3 = 0, z2,y + z22 + d1 z1 + d2 z2 + d3 = 0. Liouvillian solutions determined by first order Loewy components of (6.1). 2 2 These results show that for an equation in symmetry classes S2,2 , S3,3 or a canonical form may always be obtained from its Janet basis coefficients. 2 2 If it belongs to symmetry class S22 , S3,1 , S3,2 or S82 a canonical form may be obtained for large classes of transformation functions. An equation in symmetry class S12 in general may not be transformed to canonical form. The details of these proceedings will be described now for the various symmetry classes one after another, proofs are given for the individual cases. At first the corresponding structure invariance groups will be determined, and after that its differential invariants up to second order. As usual the notation ∆ = σx ρy − σy ρx is applied. Canonical Forms and Structure Invariance Groups. In general the canonical forms of the invariant equations for the various symmetry classes are not unique. For the special case of quasilinear equations this freedom is completely described in Theorem 6.4 below. The structure invariance groups given there have an important meaning for the respective differential equation. In the first place they describe the degree of arbitrariness for the transformation functions to canonical form. This is an important information for solving the systems of pde’s describing these transformations as it will be seen later on in this chapter. Secondly, knowing the freedom that is allowed for the canonical form is a necessary prerequirement in order to obtain definite statements on the existence of exact solutions and for designing solution algorithms. 2 S3,4

Theorem 6.4 In canonical variables u and v ≡ v(u) the second order quasilinear equations with nontrivial symmetries have the following canonical forms and corresponding structure invariance groups; f , g and r are undetermined functions of their respective arguments, a, c and ai for all i are constants. One-parameter symmetry group S12 : v 00 + r(u, v 0 ) = 0 allows the transformations u = f (x), v = g(x) + cy.

Transformation to Canonical Form

251

Two-parameter symmetry groups 2 S2,1 : v 00 + r(v 0 ) = 0 allows u = a1 x + a2 y + a3 , v = a4 x + a5 y + a6 . 2 S2,2 : v 00 u + r(v 0 ) = 0 allows u = a1 x, v = a2 x + a3 y + a4 .

Three-parameter symmetry groups √ 2 S3,1 : v 00 (u − v) + 2v 0 (v 0 + a v 0 + 1) = 0 allows u=

a1 + (a2 − a1 a3 )y a1 + (a2 − a1 a3 )x , v= . 1 − a3 x 1 − a3 y

2 S3,2 : v 00 v 3 + a = 0, a 6= 0, allows

u=

a1 (x + a3 ) a2 y , v= . 1 + a4 (x + a3 ) 1 + a4 (x + a3 )

2 S3,3 (c): v 00 + av 0(c−2)/(c−1) = 0, c 6= 0, 1, allows u = a1 x + a2 , v = a3 y + a4 . 0

2 S3,4 : v 00 − ae−v = 0 allows u = a1 x + a2 , v = a3 x + a1 y + a4 .

Eight-parameter symmetry group S82 : v 00 = 0, projective group of the plane. A general point transformation u = σ(x, y) and v = ρ(x, y) with ρ + ρy y 0 y ≡ y(x) generates the transformation v 0 = x of the first derivative. σx + σy y 0 2 For S1 , in order to avoid any dependence on y to be generated via r(u, v 0 ), σy = ρxy = ρyy = 0 is required. This yields the above structure with Proof

v0 =

1 0 1 (g + cy 0 ), v 00 = 03 (cf 0 y 00 − cf 00 y 0 + f 0 g 00 − f 00 g 0 ). f0 f

The expression for v 00 shows that no further constraints are necessary. For the two parameter groups, any dependence on x and y in the transformed function r is avoided if x = a1 u + a2 v + a3 ,

y = a4 u + a5 v + a6

where a1 , . . . , a6 are constant. The second derivative is transformed according to a1 a5 − a2 a4 00 y 00 = v . (a1 + a2 v 0 )3 2 2 For S2,1 this assures the desired structure. For S2,2 the second derivative must be proportional to the independent variable. This requires in addition a2 = a3 = 0.

252 2 For S3,1 the transformed first derivative must be proportional to y 0 ; this requires σy = ρx = 0. The condition for the invariance of the second order invariant leads to a fairly complicated system of pde’s for σ and ρ. A Janet basis for the full system is

σy = 0, ρx = 0, ρy σx −

 ρ − σ 2 2σx 2σx2 = 0, σxx + + = 0. x−y ρ−σ x−y

The transformation functions given above, containing three constants, give its general solution. 2 For S3,2 the transformed second derivative must be proportional to y 00 and must be independent of x and y 0 . This is assured if σy = 0, ρyy = 0, 2σx ρxy − σxx ρy = 0, σx ρxx − σxx ρx = 0 ρy 00 y . In the transformed equation the coefficient of the σx2 second derivative must be independent of x and proportional to y 3 . This requires in addition holds. Then v 00 =

 ρ ρ3  σ 2  ρ ρ3  3 y y x = 0, − = 0. σx x σx y ρy ρ3 y The combined constraints may be transformed into the Janet basis σy = 0, yρy − ρ = 0, σxx ρ − 2ρx σx = 0, ρxx ρ − 2ρ2x = 0 with the general solution σ=

C1 x + C2 y , ρ= . C3 (C3 x + C4 ) C3 x + C4

By a suitable change of the integration constants Ck the above expressions for the transformation functions are obtained. 2 For the group S3,3 the transformed first derivative must be proportional 0 to y ; this requires σy = ρx = 0. The transformed second derivative cannot contain a term proportional to a power of y 0 ; this requires σxx = ρyy = 0. The general solution of these equations represent the transformation functions given in the above theorem. 2 the transformed first derivative must have the form For the group S3,4 0 −y + constant; this requires σy = 0, ρy − σx = 0, σx ρxy − σxy ρx = 0, σx ρxx − σxx ρx = 0 or the equivalent Janet basis σy = 0, σxx = 0, ρy − σx = 0, ρxx = 0 with the solution given above. Finally the structure invariance group of v 00 = 0 is obviously identical to its symmetry group S82 , i.e., the eight parameter projective group of the plane.

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For later use the Lie algebras corresponding to the two- and three-parameter symmetries of the preceding theorem are given explicitly next. Corollary 6.1 Two-parameter symmetry groups 2 S2,1 : {∂x , ∂y , x∂x , y∂x , x∂y , y∂y }.

2 S2,2 : {∂y , x∂x , x∂y , y∂y }

Three-parameter symmetry groups 2 S3,1 : {∂x + ∂y , x∂x + y∂y , x2 ∂x + y 2 ∂y }. 2 S3,3 (c): {∂x , ∂y , x∂x , y∂y }.

2 S3,2 : {∂x , x∂x , y∂y , x2 ∂x + xy∂y }.

2 S3,4 : {∂x , ∂y , x∂x + y∂y , x∂y }.

The proof is a straightforward application of the definition of the infinitesimal generators given in Chapter 3. A symmetry class may comprise only a single equivalence class. This is true e. g. for S82 . Because its canonical representative v 00 = 0 does not contain any undetermined function or parameter, this symmetry class is at the same time an equivalence class as has been mentioned before. In general, however, a symmetry class may comprise more than a single equivalence class. In these cases there remains the question of how the individual equivalence classes contained in it may be uniquely parametrized. For the symmetry classes corresponding to three-parameter groups the answer is given next. 2 Corollary 6.2 The symmetry class S3,1 decomposes into equivalence classes which are uniquely determined by the value of the parameter a in the 2 2 canonical form equation. The remaining symmetry classes S3,2 , S3,3 (c) and 2 S3,4 comprise a single equivalence class. 2 Proof The structure invariance group of S3,1 is identical to the symmetry group. Therefore it cannot transform equations with different values of a into each other. In the remaining cases, canonical equations with parameter values a and b are transformed into each other if the group parameters of the respective structure invariance groups are determined by

 a  b  ac1 1/(c−1) b 1 b  a1 2 3 = , and = or = exp − . a a a3 a a1 a1 a22 That is, in all three cases there is a one-parameter set of transformations by which this transformation may be achieved. For two-parameter symmetries the situation is more complicated. A partial answer may be obtained from the following example dealing with symmetry 2 class S2,2 . Example 6.3 Applying a general transformation of the structure invari2 ance group to the canonical form uv 00 +r(v 0 ) = 0 of symmetry class S2,2 yields

254 a2 + a3 y 0 ). Its action consists essentially xy 00 + s(y 0 ) = 0 with s(y 0 ) = a13 r( a a1 1 of a linear transformation of the argument of r. Example 6.4 Applying a general transformation of the structure invari2 ance group to the canonical form v 00 v 3 + a = 0 for S3,2 symmetry yields 2 a y 00 y 3 + b = 0 with b = a 14 , i.e., any value for the constant b may be achieved, a2 e. g. b = 1. The parameters and undetermined functions occurring in the canoncial forms of Theorem 6.4 have to obey certain constraints in order to guarantee the respective symmetry type. They are a consequence of the definition that the symmetry type of an equation is determined by the maximal group leaving it invariant. Obvious examples of these constraints are a 6= 0 in the dr(v 0 ) 2 2 2 canonical forms of S3,2 , S3,3 and S3,4 6= 0 in the canonical forms , or dv 0 2 2 of S2,1 and S2,2 assuring that the projective symmetry is excluded. For the applications in this book, however, this has no importance because the symmetry type of an equation is determined by its Janet basis for the symmetries which always gives the correct answer for a given equation in actual variables. These questions are discussed in more detail in Exercise 6.1 The discussion in the remaining part of this chapter is based on a series of articles by Lie [109], especially part III. As Lie mentions at the beginning of page 371, he outlines the ample calculations only schematically. Here all necessary steps and case distinctions are described in full detail such that the theorems given below readily may be applied for solving concrete examples. The differential equations satisfied by the transformation functions to canonical form have to obey severe constraints which are a consequence of the following result due to Lie [109], part III, page 379ff. Theorem 6.5 (Lie 1883) The differential equations satisfied by the transformation functions to canonical form allow the structure invariance group of the respective symmetry class as a symmetry group. Lie’s proof uses the fact that the determining systems for the actual differential equation and the corresponding equation in canonical form are related by the desired transformation functions as well. Knowing their symmetry group and its differential invariants yields a better understanding of their structure. On page 382, part III of [109] the following remarkable result is proved. Theorem 6.6 (Lie 1883) Let Ik be the invariants for a particular symmetry type, and σ(x, y) and ρ(x, y) be the transformation functions to canonical form. There are relations of the form Ik = Bk (x, y) with known functions Bk .

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255

After these preliminary remarks the equations determining the transformation functions are determined explicitly. One-Parameter Symmetry. This is the simplest type of nontrivial invariance that may occur for any ode. The freedom involved in the canonical form transformation, i.e., two unspecified functions of a single variable and a constant, correspond to the respective quantities generating the structure invariance group. Theorem 6.7 If a second order ode belongs to symmetry class S12 , two types of Janet bases may occur; f , g, h and k are undetermined functions of its argument.
R (2,2) i) Janet basis of type J1,1 : σ = f (x), ρ = exp F (x, y) dy +g(x) where H F (x, y) = a(x, y)dx + b(x, y)dy. (2,2)

ii) Janet basis of type J1,2 : σ = h(φ) where φ(x, y) = C is the solution
R of y 0 + a = 0, Hρ = C exp G(x, y) dx + k(y) where C is a constant, and G(x, y) = b(x, y)dx + c(x, y)dy. Proof

In case i) the Janet basis ξ = 0, ηx + (log ρy )x η = 0, ηy + (log ρy )y η = 0

yields the relations (log ρy )x = a, (log ρy )y = b and σy = 0 with the solution given above. In case ii) the Janet basis   ∆  ∆  σx ξ = 0, ξy + log ξ=0 ξ = 0, ξx + log η+ σy σy x σy y leads to the system   σx ∆  ∆  = a, log = b, log =c σy σy x σy y for σ and ρ. A few simplifications lead to 1 1 1 σy − σx = 0, ρy − ρx = exp G(x, y) a a a with G as defined above. By Theorem 2.19 the invariant φ(x, y) for the first equation is obtained. By Theorem 2.20 the function w(x, y, ρ) obeys 1 w + 1 eG w = 0. The variable change the homogeneous equation wy − a x ρ a y¯ = φ(x, y) yields wx − eG wρ = 0 with the solution ρ as given above. The arbitrariness due to the undetermined elements is obviously a consequence of the structure invariance group given in Theorem 6.4. The crucial step of this proceeding is to solve the ode y 0 + a(x, y) = 0 in case ii). If a solution cannot be found it fails. Consequently, the examples of

256 second order equations with a one-parameter symmetry given in the literature generate a Janet basis with a simple expression for the coefficient a(x, y), e. g. linear in y. Many equations are in the form y 00 + r(y, y 0 ) = 0. Interchanging the dependent and the independent variable with each other by x = v, y = u
and applying (B.18) yields v 00 −v 03 r u, 10 = 0. This case is discussed in more v detail in Exercise 6.4. Example 6.5 Equation 6.90 of Kamke’s collection has been shown in 5.9 (2,2) 2y to belong to symmetry class S12 . Its type J1,2 Janet basis yields a = x , 1 and c = 0. According to case ii) y 0 + 2 y = 0 is obtained with the b = −x x 1 . Special solutions solution x2 y = C. It leads to φ = x2 y and eG = C x 2 therefore are σ = x y and ρ = log x. Inverting these relations with the result x = ev and y = ue−2v leads to the canonical form v 00 + u(u + 5)v 03 − (u + 3)v 02 + 14 v 0 = 0. Example 6.6 Equation 6.98 has been considered before in Example 5.10. (2,2) Its type J1,2 Janet basis differs from the preceding case only by the value of 2y y a; now it is a = − x . A similar calculation as above yields σ = 2 , ρ = log x x and x = ev , y = ue2v with the canoncial form v 00 + 2(2u − 1)v 02 + v 0 = 0.

(6.2) (2,2)

The second equation considered in Example 5.10 had a type J1,1 Janet 1 and b = 0. Therefore case i) of the above theorem applies. basis with a = x A few simple calculations yield the general transformation σ = f (x) and 1 , C = 1 and g = 0 the ρ = Cxy + g(x). With the special choice f = x + 1 variable transformations u=

1 1−u uv , v = xy and x = , y= x+1 u 1−u

are obtained. It transforms equation (5.30) into the canonical form, too. Two-Parameter Symmetries. According to Corollary 6.1 the structure 2 invariance group for the symmetry class S2,1 is generated by the six operators {∂x , ∂y , x∂x , y∂x , x∂y , y∂y }. Its invariants up to second order are given next. Lemma 6.1 The pde’s for the canonical form transformations σ(x, y) and 2 ρ(x, y) for the symmetry class S2,1 allow the following invariants w.r.t. the invariance group generated by {∂σ , ∂ρ , σ∂σ , ρ∂σ , σ∂ρ , ρ∂ρ }. There are no first order invariants, and there are six second order invariants σ ρ −σ ρ σ ρ −σ ρ σ ρ −σ ρ I1 = σy ρyy − σyyρ y , I2 = σx ρyy − σyyρ x , I3 = σy ρxy − σxyρ y , x y y x x y y x x y y x σ ρ −σ ρ σ ρ −σ ρ σ ρ −σ ρ I4 = σy ρxx − σxxρ y , I5 = σx ρxy − σxyρ x , I6 = σx ρxx − σxxρ x . x y y x x y y x x y y x

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257

Proof The system of pde’s determining any invariant Φ of order not higher than two comprises Φσ = Φρ = 0 and σx Φσx + σy Φσy + σxx Φσxx + σxy Φσxy + σyy Φσyy = 0, ρx Φσx + ρy Φσy + ρxx Φσxx + ρxy Φσxy + ρyy Φσyy = 0, σx Φρx + σy Φρy + σxx Φρxx + σxy Φρxy + σyy Φρyy = 0, ρx Φρx + ρy Φρy + ρxx Φρxx + ρxy Φρxy + ρyy Φρyy = 0. The subsystem containing only derivatives w.r.t. σ, ρ and its first order derivatives contains six equations in six indeterminates with a nonvanishing coefficient determinant σx ρy − σy ρx . Consequently, a first order invariant does not exist. The full system for the second order invariants comprises six equations in twelve indeterminates σ, σx , σy , σxx , σxy and σyy , and the corresponding derivatives of ρ, i.e., six second order invariants may exist. They may be obtained by the methods described in Chapter 2. (2,2)

Theorem 6.8 If a second order ode has a Janet basis of type J2,3 and 2 a1 = b2 and c1 = d2 , it belongs to symmetry class S2,1 . The transformation function σ is determined by σxx − c2 σy − a2 σx = 0, σxy − c1 σy − a1 σx = 0, σyy − d1 σy − b1 σx = 0. (6.3) There is an identical set of equations for ρ. The general solution of the full system contains six constants. Proof From the Janet basis given in equation (11) of Schwarz [165], the coefficients a1 , a2 , . . . , d2 may be expressed in terms of the transformation functions σ and ρ as σy ρxy − σxy ρy = a1 ∆, σy ρyy − σyy ρy = b1 ∆, σx ρxy − σxy ρx = −c1 ∆, σx ρyy − σyy ρx = −d1 ∆,

σy ρxx − σxx ρy = a2 ∆, σy ρxy − σxy ρy = b2 ∆, σx ρxx − σxx ρx = −c2 ∆, σx ρxy − σxy ρx = −d2 ∆

(6.4)

where as usual ∆ = σx ρy − σy ρx . This system may be transformed into a Janet basis in grlex term ordering with ρ > σ > d2 > d1 > . . . > a2 > a1 and y > x with the result b2 − a1 = 0, d2 − c1 = 0, a2,y − a1,x + c2 b1 − c1 a1 = 0, b1,x − a1,y + d1 a1 − b1 c1 + b1 a2 − a21 = 0, c2,y − c1,x + d1 c2 − c2 a1 − c21 + c1 a2 = 0, d1,x − c1,y + b1 c2 − c1 a1 = 0, σxx − c2 σy − a2 σx = 0, σxy − c1 σy − a1 σx = 0, σyy − d1 σy − b1 σx = 0, ρxx − c2 ρy − a2 ρx = 0, ρxy − c1 ρy − a1 ρx = 0, ρyy − d1 ρy − b1 ρx = 0. (6.5)

258 The lower equations not involving σ and ρ represent the coherence conditions for the Janet basis coefficients. The upper half of this Janet basis represents the two identical linear systems for σ and ρ respectively as given above. Example 6.7 The symmetry type of equation 6.227 has been identified as (1,2) 2 S2,1 in Example 5.11. By Theorem 6.8, case i) the type J3,2 Janet basis for the transformation function σ is 

σxx +

1 1 σx = 0, σxy = 0, σyy + σy = 0 , x y

and an identical one for ρ. It is not completely reducible; its Loewy factor (2,2) comprises only {σx , σy }. Dividing it out yields a type J2,3 Janet basis for the quotient in terms of σ1 ≡ σx and σ2 ≡ σy . It is completely reducible and (2,2) (2,2) decomposes into type J1,1 and type J1,2 Janet bases according to 

1 σ = 0, σ = 0, σ = 0, σ + 1 σ = 0 σ1,x + x 1,y 2,x 2,y 1 y 2  = Lclm σ1 = 0, σ2,x = 0, σ2,y + y1 σ2 = 0 , 
1 σ2 + x y σ1 = 0, σ1,x + x σ1 = 0, σ1,y = 0

1, σ = −1, from which the special solutions σ1 = 0, σ2 = y1 and σ1 = x 2 y x and finally σ = log y and σ = log y are obtained. A possible choice of the transformation functions to new variables u ≡ σ and v ≡ ρ therefore is u+v σ = log y and ρ = log x , y = eu . It yields the y with the inverse x = e 03 02 0 canonical form v 00 + v + 5v 0+ 8v + 4 = 0. v As has been shown in Theorem 6.4, the canonical form for the symmetry classes corresponding to two-parameter groups is highly non-unique. This arbitrariness is discussed next.

Example 6.8 The preceding example is continued. A fundamental system for the transformation functions σ and ρ has been shown to be {1, log x, log y}. Any linear combination leading to a nonvanishing functional determinant is appropriate, although it is not a priori clear whether there is a particularly favorable one for the subsequent solution procedure. The linear combinations u = c1 log x + c2 log y, v = c3 log x + c4 log y with c1 c4 − c2 c3 6= 0 lead to the most general transformation eu = xc1 y c2 and ev = xc3 y c4 . The special v 0 (v 0 + 1)2 choice c1 = c4 = 0, c2 = c3 = 1 leads to v 00 + = 0 for which r(v 0 ) v0 − 1 decomposes into linear factors. The transformation between the two canonical forms is discussed in Exercise 6.2. 2 The structure invariance group for the symmetry class S2,2 is generated by the four operators {∂y , x∂x , x∂y , y∂y } according to Corollary 6.1. Its invariants are similar to the preceding case.

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259

Lemma 6.2 The pde’s for the canonical form transformations σ(x, y) and 2 ρ(x, y) for the symmetry class S2,2 allow the following invariants w.r.t. the invariance group that is generated by {∂ρ , σ∂σ , σ∂ρ , ρ∂ρ }. There are two first σ order invariants K1 = σσx and K2 = σy . If σx 6= 0, ρx 6= 0 and ρy 6= 0, there are the same six second order invariants that have been determined in Lemma 6.1 Proof The system of pde’s determining the invariants Φ of order not higher than two comprises Φρ = 0 and σΦσ + σx Φσx + σy Φσy + σxx Φσxx + σxy Φσxy + σyy Φσyy = 0, σx Φρx + σy Φρy + σxx Φρxx + σxy Φρxy + σyy Φρyy = 0, ρx Φρx + ρy Φρy + ρxx Φρxx + ρxy Φρxy + ρyy Φρyy = 0. The subsystem containing only derivatives w.r.t. σ, ρ and its first derivatives comprises four independent equations in six variables; i.e., there are two first order invariants. The four independent equations in twelve variables of the full system allow 12-4-2=6 second order invariants. In Exercise 6.3 the Janet basis coefficients of the two parameter symmetries will be determined in terms of the respective invariants. (2,2)

Theorem 6.9 If a second order ode has a Janet basis of type J2,3 and 2 belongs to symmetry class S2,2 , the transformation function σ is hI i σ = exp (c1 − d2 )dx + (b2 − a1 )dy . (6.6) For the other transformation function ρ two alternatives may occur. a) If c1 6= d2 , ρ is explicitly given by I I ρ1 c2 dx + c1 dy c2 dx + d2 dy −σ ρ = ρ1 d2 − c1 σ d2 − c1

(6.7)

where ρ1 = exp

hI  a −b   a −b  i 1 2 1 2 c2 + c1 dx+ c1 + b2 − a1 + d1 dy . c1 − d2 c1 − d2

b) If c1 = d2 , ρ is explicitly given by I I a1 dx − b1 dy ρ1 b2 dx + b1 dy ρ = ρ1 −σ a1 − b2 σ a1 − b2 where ρ1 = exp

I

 a2 dx + b2 dy .

(6.8)

260 Proof From the Janet basis given in equation (10) of Schwarz [165], the coefficients a1 , a2 , . . . , d2 may be expressed in terms of the transformation functions σ and ρ as (σy ρxy − σxy ρy )σ + σy ∆ = −a1 σ∆, σy ρyy − σyy ρy = −b1 ∆, σx ρxy − σxy ρx = c1 ∆, (σx ρyy − σyy ρx )σ − σy ∆ = d1 σ∆,

(σy ρxx − σxx ρy )σ + σx ∆ = −a2 σ∆, σy ρxy − σxy ρy = −b2 ∆, σx ρxx − σxx ρx = c2 ∆, (σx ρxy − σxy ρx )σ − σx ∆ = d2 σ∆. (6.9) In the same term ordering as system (6.5) above the Janet basis a2,y − a1,x + c2 b1 − c1 b2 = 0, b1,x − a1,y + a1 d1 − b1 c1 − a1 b2 + a2 b1 = 0, b2,x − a1,x + a1 d2 − c1 b2 = 0, b2,y − a1,y + b1 d2 − b2 d1 + a1 d1 − c1 b2 = 0, c2,y − c1,x − c1 d2 + c2 d1 − a1 c2 + c1 a2 = 0, d1,x − c1,y − a1 d2 + b1 c2 = 0, d2,x − c1,x − a2 d2 + b2 c2 − a1 c2 + c1 a2 = 0, d2,y − c1,y − a1 d2 + c1 b2 = 0, σx + (d2 − c1 )σ = 0, σy + (a1 − b2 )σ = 0, ρxx − c2 ρy + (d2 − c1 − a2 )ρx = 0, ρxy − c1 ρy − b2 ρx = 0, ρyy + (a1 − d1 − b2 )ρy − cρx = 0 is obtained. The eight lowest equations, i.e., the first four lines, are the coherence conditions for the coefficients. The two subsequent equations determine σ in terms of the path integral (6.6). The three remaining equations for ρ allow the solutions ρ = 1 and ρ = σ. This follows from the fact that they may be reduced to zero by both {ρx , ρy } and also by {ρx + (d2 − c1 )ρ, ρy + (a1 − b2 )ρ}, i.e., the same system that determines σ. If c1 6= d2 , dividing out the least common left multiple
Lclm {ρx , ρy }, {ρx + (d2 − c1 )ρ, ρy + (a1 − b2 )ρ} n   o = ρ1 ≡ ρy + a1 − b2 ρx , ρ2 ≡ ρxx + a1 − b2 c2 + d2 − c1 − a2 ρx c1 − d2 c1 − d2 (6.10) yields the exact quotient n  a −b   a −b  o 1 2 1 2 c2 +c1 ρ1 , ρ1,y − c1 +b2 −a1 +d1 ρ1 ρ2 −c2 ρ1 , ρ1,x − c1 − d2 c1 − d2 from which ρ1 and ρ2 may be be determined. Consequently, the second line of (6.10) forms an inhomogeneous system of pde’s with known solution of the homogeneous part and known right hand sides. It has been shown in Excercise 5 of Chapter 2, see also Appendix A, how to express a special solution in terms of integrals. Applying the results obtained there leads to the expression (6.7). If c1 = d2 it follows c1 = c2 = d2 = 0. The least common left multiple now is
Lclm {ρx , ρy }, {ρx , ρy + (a1 − b2 ρ}) = (6.11) {ρ1 ≡ ρx , ρ2 ≡ ρyy + (a1 − b2 − d1 )ρy }

Transformation to Canonical Form

261

and yields the exact quotient {ρ2 − b1 ρ1 , ρ1,x − a2 ρ1 , ρ1,y − b2 ρ1 }. Applying the results of Exercise 5 in Chapter 2 leads to the representation (6.8). 2 This result for the symmetry class S2,2 is more explicit than for symmetry 2 class S2,1 because the transformation functions are obtained in closed form without solving any equation. Consequently, whenever a second order quasilinear equation allows this symmetry type, it is guaranteed that it may be transformed into canonical form by a Liouvillian transformation.

Example 6.9 It has been shown in Example 5.12 that Equation 6.159 2 of Kamke’s collection belongs to symmetry class S2,2 . The integral (6.6) of 1 Theorem 6.9 yields σ = √ . Because c1 = d2 = 0, the integral (6.8) yields y ρ = x. The inverse of this transformation x = v, y = 12 yields the canonical u form v 00 u − 23 v 0 (v 02 − 1) = 0. Three-Parameter Symmetries. The general proceeding in this subsection is the same as in the preceding one, i.e., at first the invariants of the structure invariance groups for the various symmetry classes are determined. Subsequently the equations for the transformation functions to canonical form are obtained. They turn out to be considerably more complicated than for the two-parameter symmetries. Lemma 6.3 The pde’s for the canonical form transformations σ(x, y) and 2 ρ(x, y) for the symmetry type S3,1 allow the following invariants w.r.t. the invariance group that is generated by {∂σ + ∂ρ , σ∂σ + ρ∂ρ , σ 2 ∂σ + ρ2 ∂ρ }. If σ ρ σx 6= 0 and ρx 6= 0 there are three first order invariants K1 = σxy , K2 = ρxy σx ρx and K3 = , and six second order invariants (ρ − σ)2 I1 =

σxx 2σx σxy 2σy σyy 2σy + , I2 = + , I3 = + , σx ρ−σ σx ρ−σ σy ρ−σ

I4 =

ρxx 2ρx ρxy 2ρy ρyy 2ρy − , I5 = − , I6 = − . ρx ρ−σ ρx ρ−σ ρy ρ−σ

Proof The system of pde’s determining any invariant Φ of order not higher than two comprises Φσ + Φρ = 0 and σΦσ + ρΦρ + σx Φσx + σy Φσy + ρx Φρx + ρy Φρy +σxx Φσxx + σxy Φσxy + σyy Φσyy + ρxx Φρxx + ρxy Φρxy + ρyy Φρyy = 0, σ 2 Φσ + ρ2 Φρ + 2σσx Φσx + 2σσy Φσy + 2ρρx Φρx + 2ρρy Φρy +2(σσxx + σx2 )Φσxx + 2(σσxy + σx σy )Φσxy + 2(σσyy + σy2 )Φσyy +2(ρρxx + ρ2x )Φρxx + 2(ρρxy + ρx ρy )Φρxy + 2(ρρyy + ρ2y )Φρyy = 0.

262 The subsystem containing only derivatives w.r.t. σ, ρ and its first derivatives comprises three independent equations in six variables, i.e., there are three first order invariants. The three independent equations in twelve variables involving second derivatives allow 12-3-3=6 second order invariants. 2 Theorem 6.10 If a second order ode belongs to symmetry class S3,1 two types of Janet bases may occur. The transformation functions σ and ρ are determined by the following systems of equations. (2,2)

a) Janet basis type J3,6 : Three cases have to be distinguished. If b1 6= 0 define z 2 ≡ a1 b1 + 1 and   1 P (x, y) ≡ (z + 1)(a1 b1 + 1)b3 − z(a3 b1 + c3 − b2 )b1 , (a1 b1 + 1)b1  Q(x, y) ≡ b12 z 2 (a3,x − a1 d3 − d2 ) + a3 (a3 b1 + 2c3 − 3b2 ) 4z  −a1 (c3 − b2 )2 − a1 a3 b1 b2 . (6.12) R is determined from Rx + R2 − P R + Q = 0, h   i Ry + z + 1 R 2 − z + 1 P + z + 1 R + z − 1 Q = 0. b1 b1 b1 b1 x

(6.13)

The system for σ and ρ is 1 σ + σ = 0. σy − z + 1 σx = 0, σxx + (2R − P )σx = 0, ρ = R x b1 (6.14) If b1 = 0 define P ≡

1 (a3,x − a1 d3 − a1 c23 + 2a3 c3 − d2 ), Q ≡ a1 c3 + c2 − a3 . (6.15) 2

R is either determined from Rx + P = 0, Ry + R2 − QR − 21 a1 P = 0

(6.16)

and the system for σ(x, y) and ρ(x, y) is σx = 0, σyy + (2R − Q)σy = 0, ρ =

1 σy + σ. R

(6.17)

Or R is determined from Rx + R2 − c3 R = 0, Ry + 12 a1 R2 − (a1 c3 − a3 )R + P = 0

(6.18)

and the system for σ and ρ is 1 σ + σ = 0. σy + 21 a1 σx = 0, σxx + (2R − c3 )σx = 0, ρ = R x

(6.19)

Transformation to Canonical Form

263

(2,2)

b) Janet basis type J3,7 : Defining b1 and B by  I cx + a3 c  2 dy , b1 ≡ −c , B ≡ exp 2 a3 dx + c2 at first R and S may be determined from Rx + R2 − (a2 c + a3 )R + B = 0, c − (a2 c + a3 )c Ry − 1c R2 − x R − 1c B = 0, c2 Sx − S 2 + (a2 c − a3 )S − B = 0,

(6.20)

c + (a2 c − a3 )c S + 1c B = 0. Sy + 1c S 2 − x c2 Then the system for σ and ρ is σy − 1c σx = 0, σxx + (2R − a2 c − a3 )σx = 0, ρy + 1c ρx = 0, ρxx − (2S − a2 c + a3 )ρx = 0.

(6.21)

Proof At first case a) will be considered. If b1 6= 0, the Janet basis for σ and ρ in grlex ordering with ρ > σ, y > x is σy − z + 1 σx = 0, ρy + z − 1 ρx = 0, b1 b1 σxx (ρ − σ) + 2σx2 − P σx (ρ − σ) = 0, ρx σx − Q(ρ − σ)2 = 0. The new function R ≡ ρ σ−x σ is introduced with the result σx − R(ρ − σ) = 0, σy − z + 1 R(ρ − σ) = 0, b1 Q Q z ρx − P (ρ − σ) = 0, ρy − − 1 R (ρ − σ) = 0. b1 P , Q and z are defined as above. The integrability conditions of this system are the two first equations of (6.14). If a lex term order with ρ > σ > R is applied, the complete system (6.14) is obtained. If b1 = 0, σx = 0, ρx 6= 0, the Janet basis for σ and ρ in grlex term ordering with ρ > σ, x > y is σx = 0, σyy (ρ − σ) + 2σy2 − Qσy (ρ − σ) = 0, ρx σy − P (ρ − σ)2 = 0, ρy σy + 21 a1 P (ρ − σ)2 = 0 σ where P and Q are defined by (6.15). The new function R = ρ −y σ is introduced. It yields the system σx = 0, σyy + (2R − Q)σy = 0, ρy R + 21 a1 P (ρ − σ) = 0, ρx R − P (ρ − σ) = 0.

264 Its integrability conditions are the equations (6.16). In order to show this, the relations Qx + 2P = 0, Px = c3 P and Py = c2 P are useful. If b1 = 0, σx 6= 0 and ρx = 0, the Janet basis for σ and ρ in the same term ordering as above is σy + 21 a1 σx = 0, σxx (ρ − σ) + 2σx2 − c3 σx (ρ − σ) = 0, ρx = 0,

ρy σx − P (ρ − σ)2 = 0.

Introducing now R = ρ σ−x σ as a new function, the system σx − R(ρ − σ) = 0, σy + 21 a1 R(ρ − σ) = 0, ρx = 0, ρy R − P (ρ − σ) = 0 for σ and ρ is obtained. A similar reasoning to the above leads to equations (6.18) and (6.19). (2,2) In case b), the type J3,7 Janet basis for σ and ρ is σy − 1c σx = 0, ρy + 1c ρx = 0, σxx (ρ − σ) + 2σx2 − (a2 c + a3 )σx (ρ − σ) = 0, ρxx (ρ − σ) − 2ρ2x + (a2 c − a3 )ρx (ρ − σ) = 0. ρ Defining R ≡ ρ σ−x σ and S ≡ ρ −x σ leads to the equations Rx + R2 + RS − (a2 c + a3 )R = 0, Ry − 1c R2 − 1c RS − 12 [cx − (a2 c + a3 )c]R = 0, c Sx − S 2 − RS + (a2 c − a3 )S = 0, Sy + 1c S 2 + 1c RS − 12 [cx + (a2 c − a3 )c]S = 0. c Combining the two equations determining the x-derivatives and the y-derivatives respectively leads to (RS)x − 2a3 RS = 0, (RS)y −

2 (cx + a3 c)RS = 0 c2

from which the above expression for B follows. Example 6.10 The equation considered in Example 5.13 in symmetry 2 is continued. Due to b1 6= 0 the first alternative of case a) in the class S3,1 above theorem applies. The Riccati system (6.13) has the form

Rx + R 2 +

2x

2 y

2 R −  2 2 2 = 0, x − y y   2 1 R=0 2 Ry + x R + − y y 2 y− 2 x 2

x −

Transformation to Canonical Form

265

y −xy + C + xy . Substituting the special x2 y − 2 corresponding to C → ∞ into (6.14) yields

with the rational solution R = solution

−xy x2 y − 2

σxx = 0, σy −

2 x σx = 0, ρ + (x − )σx − σ = 0. y xy

The Janet basis for σ is completely reducible, it may be written as   1 1 Lclm σx − σ, σy − σ , {σx , σy } . x y 2 . The corresponding transforThe non-constant solution σ = xy yields ρ = x mation to canonical variables u and v is x = v2 , y = 12 uv. Substitution into the original equation finally yields the equation in canonical form v 002 (u − v)2 + 4v 00 v 0 (v 0 + 1)(u − v) + 4v 04 + 4v 03 + 4v 02 = 0. Lemma 6.4 The pde’s for the canonical form transformations σ(x, y) and 2 ρ(x, y) for the symmetry class S3,2 allow the following invariants w.r.t. the invariance group that is generated by {∂σ , σ∂σ , ρ∂ρ , σ 2 ∂σ + σρ∂ρ }. If σx 6= 0 σ ρ ρ σ and ρx 6= 0 there are two first order invariants K1 = σxy , K2 = ρy − ρx σxy , and six second order invariants σxx σxy σyy ρx ρx ρy I1 = − 2 , I2 = − 2 , I3 = −2 , σx ρ σy ρ σy ρ I4 =

ρxx ρxy ρyy ρx σxx 1 ρx σxy ρy σyy 1 ρy σxy − − − , I5 = − , I6 = . ρ ρ σx ρ 2 ρ σx 2 ρ σy ρ ρ σy

Proof The system of pde’s determining any invariant Φ of order not higher than two comprises Φσ = 0 and σx Φσx + σy Φσy + σxx Φσxx + σxy Φσxy + σyy Φσyy = 0, ρΦρ + ρx Φρx + ρy Φρy + ρxx Φρxx + ρxy Φρxy + ρyy Φρyy = 0, σρΦρ + 2σσx Φσx + 2σσy Φσy + (ρσx + σρx )Φρx + (ρσy + σρy )Φρy +2(σσxx + σx2 )Φσxx + 2(σσxy + σx σy )Φσxy + 2(σσyy + σy2 )Φσyy +(ρσxx + σρxx + 2σx ρx )Φρxx + (ρσxy + σρxy + σx ρy + σy ρx )Φρxy +(ρσyy + σρyy + 2σy ρy )Φρyy = 0. The number of invariants is determined in the same way as in the proof of Lemma 6.2. The four parameters of the structure invariance group for the symmetry 2 class S3,2 are obtained by combining the integration constant of a second order Riccati-like system and a second order Janet basis for σ or ρ respectively.

266 2 Theorem 6.11 If a second order ode belongs to symmetry class S3,2 , three types of Janet bases may occur. The transformation functions σ and ρ are determined by the following systems of equations. (2,2)

a) Janet basis of type J3,4 . Rxx + Rx2 − a3 Rx − 14 (c3,x + c23 − a3 c3 − d2 ) = 0, Ry − 21 a2 = 0, ρx + Rx ρ = 0, ρy + Ry ρ = 0, σy = 0, σxx − (2Rx − a3 )σx = 0. (6.22) (2,2)

b) Janet basis of type J3,7 . Ryy + Ry2 − c2 Ry − 41 (a2,y + a22 − a2 c2 − d3 ) = 0, Rx − 12 c3 = 0, ρx + Rx ρ = 0, ρy + Ry ρ = 0, σx = 0, σyy + (2Ry − c2 )σy = 0. (6.23) (2,2)

c) Janet basis of type J3,6 . Q Rxx + Rx2 + a1 b3 Rx − P = 0, Ry + a1 Rx − 21 P = 0, ρx + Rx ρ = 0, ρy + Ry ρ = 0, σy + a1 σx = 0, σxx + (2Rx + a1 b3 )σx = 0 (6.24) ∆ where P (x, y) ≡ a1 c3 − a1 b2 − a3 ≡ −2 σx ρ 6= 0 and Q(x, y) =

1 4

 (b1 a3 + c3 − b2 )(a3,x − d2 ) + a1 (c3 − b2 )3  −d3 P (x, y) + 2b3 P (x, y)2 + a3 (b2 c3 − a3 b1 b2 − c23 ) .

Proof The proof will be given in detail for case a) for Janet basis type (2,2) J3,4 . If its coefficients are compared to the corresponding expressions in terms of the transformation functions (see also Theorem 5.8), the following system of pde’s is obtained. (log ρ2 )y = a2 , 



log σx2 ρ

 x

= a3 ,



ρ log ρy

 y

= c1 , c2 = 0,



ρ log ρy

 x

= c3 ,

  ρ2  x log σyx = d1 , (log ρx )y (log σx )x − (log ρxx )y + 3(log ρ)x log σ ρx x = d 2 , h      i ρx ρxx σx σx log +3 log log ρy σxx x ρx x ρ x = d3 .

Transformation to Canonical Form

267

From this system of equations a Janet basis in total degree, then lexicographic term order ρ > σ > d3 > d2 > . . . > a3 > a2 is generated. d1 − 2c3 − a3 = 0, a2,x − a2 c3 = 0, a2,y − a2 c2 = 0, a3,y − c3 a2 = 0, c3,y − c2,x = 0, d3,y − d2,x + d3 c2 − 2d3 a2 − d2 c3 + 2d2 a3 = 0, c2,xx − d2,y − 2c3,x a2 + 2c2,x c3 − c2,x a3 − 3c23 a2 + 2a2 a3 c3 = 0, c3,xx − d2,x + 2c3,x c3 − 3c3,x a3 − a3,x c3 − 2d3 a2 + 2d2 a3 − 2c23 a3 + 2c3 a23 = 0, σy = 0, σxx ρ − 2σx ρx − σx ρa3 = 0, ρy +

1 2 ρa2

= 0, ρxx ρ − 2ρ2x − ρx ρa3 + 41 ρ2 (c3,x − d2 + c23 − a3 c3 ) = 0.

Introducing the new function R ≡ − log ρ into the last four equations yields system (6.22). For case b) the four highest equations of the Janet basis for the full system expressing σ and ρ in terms of the coefficients a2 , . . . , d2 , d3 are σx = 0, ρx + 21 c3 ρ = 0, σyy ρ − 2ρy σy − c2 σy ρ = 0, ρyy ρ − 2ρ2y − ρy ρc2 + 14 (a2,y + a22 − d3 − a2 c2 )ρ2 = 0. Substituting R ≡ − log ρ yields (6.24). Finally for case c) a fairly complicated system of pde’s is obtained for σ and ρ. Introducting again R ≡ − log ρ, it may be written in the form (6.23). The equations for R in the systems (6.22), (6.23) and (6.24) are of the form (2.58) on page 89. The solution procedure described there is always applied. Example 6.11 The symmetry class of equation 6.133 of Kamke’s collection 2 has been identified as S3,2 in Example 5.14. Because its Janet basis is of type (2,2)

J3,6 , case c) of the above theorem applies. The system (6.24) is 1

1 R − 1 4 Rxx + Rx2 + x + y x (x + y)2 = 0, Ry + Rx + x + y = 0. Solving the first equation leads to the system Rx =

1 1 − 2 , x−y+C x+y

Ry = −

1 1 − 2 x−y+C x+y

for R with the integration constant C ≡ C(y). Because the equation for C(y) obtained from (2.60) is linear, the special solution corresponding to C → ∞ may be chosen and the system for σ becomes {σy + σx , σxx }. It is completely reducible and may be written as 
Lclm σx − x − y1+ C σ, σy + x − y1+ C σ = 1

Lclm



1

 1 σ, σ − 1 σ = 0. σx , σy }, σx + x − y y x−y 

268 The two components of the latter representation correspond to the values C1 → ∞ and C1 = 0 respectively. A special solution is σ = x − y. To 1 . the chosen value of C there corresponds the system Rx = Ry = 2(x + y) √ 1 A special solution is R = 2 log (x + y) and consequently ρ = x + y. The inverse of the transformation √ u = x − y, v = x + y is x = 12 (v 2 + u), y = 12 (v 2 − u), yields the canonical form v 00 v 3 + 14 = 0 for equation 6.133 of Kamke’s collection. The transformations of the structure invariance group may be applied to generate any value for the constant term. Choosing a1 = 2, a2 = 1 and a3 = a4 for example yields v 00 v 3 + 1 = 0. 2 2 For both symmetry classes S3,3 (c) and S3,4 the four-parameter structure invariance originates from the second order Janet bases for the transformation functions σ and ρ as it is shown in the next lemma. Moreover, the same 2 remarks apply as for symmetry class S2,1 on page 261, i.e., no equations have to be solved in order to determine the transformation functions.

Lemma 6.5 The pde’s for the canonical form transformations σ(x, y) and 2 ρ(x, y) for symmetry class S3,3 (c) allow the following invariants w.r.t. the invariance group that is generated by {∂σ , ∂ρ , σ∂σ , ρ∂ρ }. If all first derivatives σ ρ are different from zero there are two first order invariants K1 = σxy , K2 = ρxy and six second order invariants σxy σyy ρxx ρxy ρyy σxx I1 = , I2 = , I3 = , I4 = , I5 = , I6 = . σx σx σy ρx ρx ρy Proof The system of pde’s determining any invariant Φ of order not higher than two comprises Φσ = Φρ = 0 and σx Φσx + σy Φσy + σxx Φσxx + σxy Φσxy + σyy Φσyy = 0, ρx Φρx + ρy Φρy + ρxx Φρxx + ρxy Φρxy + ρyy Φρyy = 0. The number of invariants is determined in the same way as in the proof of Lemma 6.2. 2 For the symmetry class S3,3 (c) the parameter c has been determined in Theorem 5.8 and may be considered as already known. 2 Theorem 6.12 If a second order ode belongs to symmetry class S3,3 (c) with c 6= 1, two types of Janet bases may occur. The transformation functions σ and ρ are determined by the following systems of pde’s. (2,2)

a) Janet basis of type J3,6 and σx ρy − cσy ρx 6= 0. There are two alternatives. If b1 = 0 the system is either
σx = 0, σyy − a1 c3 + c2 + ac3 σy = 0, (6.25) 1 ρy + cca − 1 ρx = 0, ρxx + cc3 ρx = 0

Transformation to Canonical Form or

σy − c a−1 1 σx = 0, σxx − d1 σx = 0, ρx = 0, ρyy − (a1 c3 + c2 + ca3 )ρy = 0.

269

(6.26)

If b1 6= 0 and c + 1 6= 0, the system is R(x, y) σy − c1 + c σx = 0, σxx + σ = 0, (c + 1)b1 T (x, y) x S(x, y) ρ =0 ρy − cc1 + 1 ρx = 0, ρxx + (c + 1)b1 T (x, y) x

(6.27)

where   R(x, y) ≡ c2 b1 (a3 b1 + c3 − b2 − a1 b3 ) − b3   +c b1 (c1 + 2)(a3 b1 − a1 b3 + c3 − b2 ) − (2c1 + 1)b3   +(c1 + 1) b1 (a3 b1 − a1 b3 + c3 − b2 ) − c1 b3 , S(x, y) ≡ c3 b1 (a3 b1 + c3 − b2 )   +c2 b1 (c1 + 2)(a3 b1 + c3 − b2 ) − (a1 b1 + 1)c1 b3   +c b1 (c1 + 1)(a3 b1 + c3 − b2 ) − (a1 b1 c1 + 1)b3 ,   T (x, y) = b1 c1 + a1 b1 + c(c + 2)(a1 b1 + 1) . (2,2)

b) Janet basis of type J3,7 and σx ρy − cσy ρx = 0. The transformation functions σ and ρ are determined by + 1 σ = 0, σ + a2 c1 − (c + 1)a3 σ = 0, σy − ccc x x xx c+1 1 a c + (c + 1)a3 ρy − c c+1 1 ρx = 0, ρxx + 2 1 c + 1 ρx = 0.

(6.28)

Proof The original expressions of the Janet basis coefficients in terms of σ and ρ are too voluminous to be given here explicitly. If they are transformed into a Janet basis in total degree, then lexicographic term order with ρ > σ and y > x, the above linear system for σ and ρ is obtained. Example 6.12 In Chapter 5, Example 5.15, two equations in symmetry 2 class S3,3 (c) have been considered. For the first equation, due to b1 = 0 the first alternative of case a) applies. It is not a priori clear which choice of c leads to the correct canonical form. For the first alternative, c = 32 and the system 2 σx = 0, σyy + σy = 0, ρy = 0, ρxx = 0 y is obtained. Its general solution is σ = Cy1 + C2 , ρ = Cx3 + C4 . The special choice C1 = C3 = 1, C2 = C4 = 0 yields the canonical form v 00 + v 04 = 0. The

270 second alternative leads to y 0 y 00 − 1 = 0. Therefore the former choice is the correct one. For the second equation c is determined by c2 − 13 6 c + 1 = 0 with the same solutions as above. The value c = 32 leads to the Janet basis σy − 12 σx = 0, σxx = 0, ρy − 31 ρx = 0, ρxx = 0. A special solution is σ = 2x+y, ρ = 3x+y. Substituting the inverse x = v −u and y = 3u − 2v into the latter equation of Example 5.15 yields v 00 + v 04 = 0, i.e., the same canonical form as the preceding equation. Lemma 6.6 The pde’s for the canonical form transformations σ(x, y) and 2 ρ(x, y) for the symmetry class S3,4 allow the following invariants w.r.t. the invariance group that is generated by {∂σ , ∂ρ , σ∂σ + ρ∂ρ , σ∂ρ }. If σx 6= 0 there σ ρ −σ ρ σ are two first order invariants K1 = σxy , K2 = x y 2 y x . If σx 6= 0 and σx σy 6= 0 there are six second order invariants I1 = I4 =

σxx σxy σyy , I2 = , I3 = , σx σx σy

σyy ρy − σy ρyy σxy ρx − σx ρxy σxx ρx − σx ρxx , I5 = , I6 = . 2 2 σx σy σx2

Proof The system of pde’s determining any invariant Φ of order not higher than two comprises Φσ = Φρ = 0 and σx Φσx + σy Φσy + ρx Φρx + ρy Φρy + σxx Φσxx + σxy Φσxy +σyy Φσyy + ρxx Φρxx + ρxy Φρxy + ρyy Φρyy = 0, σx Φρx + σy Φρy + σxx Φρxx + σxy Φρxy + σyy Φρyy = 0. The number of invariants is determined in the same way as in the proof of Lemma 6.2. 2 Theorem 6.13 If a second order ode belongs to symmetry class S3,4 two types of Janet bases may occur. (2,2)

a) If the Janet basis is of type J3,6 three cases have to be distinguished. If b1 6= 0, c1 + 1 6= 0 and T (x, y) ≡ 2a1 b1 + c1 + 1 6= 0, the transformation functions σ and ρ are determined by σy + c 2a+1 1 σx = 0, 1 ρy − c1 + 1 ρx − c1 − 1 σx = 0, 2b1 2b1

σxx + ρxx +

R(x, y) σ = 0, T (x, y) x

R(x, y) S(x, y) ρ + σ =0 T (x, y) x b1 T (x, y) x (6.29)

where R(x, y) ≡ (c1 + 1)(a3 b1 + c3 − 4) − (c1 − 1)a1 b3 , S(x, y) ≡ (c1 + 1)[(a3 b1 + c3 − b2 )b1 + b3 ] − (c1 − 3)a1 b1 b3 .

Transformation to Canonical Form

271

If b1 6= 0, c1 + 1 = T = 0 the system for σ and ρ is σy = 0, σxx − (c3 − b2 )σx = 0, f
σ = 0. ρy + 1 σx = 0, ρxx + (c3 − b2 )ρx + c3 − b2 + b1 b1 x

(6.30)

If b1 = c1 + 1 = T = 0 there is always a1 6= 0, the system for σ and ρ is σx = 0, σyy − (a1 c3 + a3 − c2 )σy = 0, a2
σ = 0. ρx + a11 σy = 0, ρyy − (a1 c3 + a3 − c2 )ρy − a1 c3 + a3 − a y 1 (6.31) (2,2)

b) If the Janet basis is of type J3,7 and ρ is

there holds c1 6= 0. The system for σ


σy − c21 σx = 0, σxx + 21 a2 c1 − a3 σx = 0,
ρy − c21 ρx − c21 σx = 0, ρxx 12 a2 c1 − a3 ρx + 12 a2 c1 σx = 0.

(6.32)

Proof At first case a) is considered. In terms of σ and ρ the coefficients a1 , b1 and c1 are a1 = −

σy2 σx2 σx σy + ∆ , b1 = , c1 = σx σ y − ∆ σx σy − ∆ σx σy − ∆

where as usual ∆ = σx ρy − σy ρx . It follows that c1 + 1 =

2σx σy ∆ 2σx σy , T (x, y) = − . σx σ y − ∆ σx σy − ∆

The three alternatives correspond to σx 6= 0 and σy 6= 0, σx 6= 0, σy = 0 and σx = 0, σy 6= 0 respectively. In the latter case there follows a1 6= 0. In x case b) there holds c1 = 2 σ σy . Consequently, c1 = 0 entails σx = 0 which is not possible due to the constraint σx σy = ∆ 6= 0, i.e., p 6= 0 is assured. Example 6.13 An equation with this symmetry type has been considered (2,2) in Example 5.16. From the type J3,6 Janet basis given there it follows a1 = 0, b1 = − x y and c1 = −1. Therefore the second alternative of case a) applies. It yields σy = 0, σxx +

1 y 1 1 σx = 0, ρy − σx = 0, ρxx − ρx + σx = 0. x x x x

A non-constant solution for σ is σ = x2 . Substituting it in the equations for ρ yields the independent solution ρ = x + y 2 .

272

The Symmetry Class of the Projective Group. The largest group of point symmetries that any second order ode may allow has been shown to be the eight-parameter projective group of R2 . Because there is no degree of freedom in the canonical form v 00 = 0 corresponding to this symmetry, its structure invariance group is identical to the symmetry group itself. The invariants of order two of the projective group and its largest subgroup are given in the following lemma. The notation for the invariants is chosen such that its relation to the differential equations to be discussed subsequently is obvious. Lemma 6.7 (Lie 1883) The projective group of the plane with generators  ∂σ , ∂ρ , σ∂ρ , ρ∂ρ , σ∂σ , ρ∂σ , σ 2 ∂σ + σρ∂ρ , σρ∂σ + ρ2 ∂ρ (6.33) does not have any first order invariants. A fundamental system of four second order invariants is σx ρyy − σyy ρx σy ρxy − σxy ρy σy ρyy − σyy ρy σx ρy − σy ρx ≡ A, σx ρy − σy ρx + 2 σx ρy − σy ρx ≡ B, σy ρxx − σxx ρy σx ρxy − σxy ρx σx ρxx − σxx ρx σx ρy − σy ρx + 2 σx ρy − σy ρx ≡ C, σx ρy − σy ρx ≡ D.

(6.34)

The largest subgroup of the projective group is generated by  ∂σ , ∂ρ , σ∂ρ , ρ∂ρ , σ∂σ , ρ∂σ , it does not have any first order invariant either. A fundamental system of six second order invariants is σx ρyy − σyy ρx σy ρxy − σxy ρy σy ρyy − σyy ρy σx ρy − σy ρx ≡ A, σx ρy − σy ρx ≡ B1 , σx ρy − σy ρx ≡ B2 , σy ρxx − σxx ρy σx ρxy − σxy ρx σx ρxx − σxx ρx σx ρy − σy ρx ≡ C1 , σx ρy − σy ρx ≡ C2 , σx ρy − σy ρx ≡ D. The proof may be found in the above quoted article by Lie [109], part III, §1. Theorem 6.14

(Lie 1883) Any second order ode

y 00 + A(x, y)y 03 + B(x, y)y 02 + C(x, y)y 0 + D(x, y) = 0

(6.35)

with projective symmetry group is similar to the equation v 00 (u) = 0. The transformation functions u = σ(x, y) and v = ρ(x, y) are solutions of σxx − Dσy + (C − 2C2 )σx = 0, σxy − C2 σy + B2 σx = 0, σyy − (B − 2B2 )σy + Aσx = 0

(6.36)

Transformation to Canonical Form

273

and an identical system for ρ such that σx ρy − σy ρx 6= 0. The coefficients B2 = b and C2 = −a are determined by the Riccati system ax + a2 + Ca − Db + Dy + BD = 0, ay + ab − 13 Bx + 23 Cy + AD = 0, bx + ab − 32 Bx + 31 Cy + AD = 0, by + b2 − Bb + Aa − Ax + AC = 0

(6.37)

of the form (2.65). Its rational solutions may be determined by the algorithm described in Chapter 2. Substituting them into (6.36) generates Janet bases of (1,2) type J3,2 for σ and ρ. They may be decomposed by the algorithms described in Chapter 2 as well. Proof A general point transformation u = σ(x, y), v = ρ(x, y) of v 00 (u) = 0 generates an equation of the form (6.35) where the coefficients A, B, C and D are given by the corresponding expressions in Lemma 6.7. These equations may be considered as a system of pde’s expressing σ and ρ in terms of the coefficients of (6.35). Furthermore, there are the relations expressing B1 , B2 , C1 and C2 from the second part of this lemma in terms of σ and ρ. By means of the obvious relations B = B1 +2B2 and C = C1 +2C2 , B1 and C1 are expressed in terms of B2 and C2 , and the resulting system is transformed into a Janet basis with lexicographical term ordering ρ > σ > B2 > C2 > A > . . . > D. The full Janet basis comprises twelve equations, the upper half of which is given by (6.36), and an identical system for ρ. The two lowest equations are the constraints for A, . . . , D of Theorem 5.11 guaranteeing the projective symmetry. In between there is the system C2,x − C22 + DB2 + CC2 − Dy − BD = 0, C2,y + B2 C2 + 31 Bx − 23 Cy − AD = 0, B2,x − B2 C2 − 23 Bx + 13 Cy + AD = 0,

(6.38)

B2,y + B22 − BB2 − AC2 − Ax + AC = 0. Combined with the two lowest equations they may be considered as coherence conditions for the linear homogeneous system (6.36). Equations (6.38) express the functions B2 and C2 in terms of the known coefficients A, B, C and D. Substituting C2 = −a and B2 = b yields the Riccati system (6.37) for a and b. The above decomposition of the coefficients of equation (6.35) into B1 , B2 , C1 and C2 is not obvious. It seems to be more natural to start with the coefficients A, B, C and D, expressed in terms of σ and ρ, compare them to the actual coefficients in terms of x and y and generate a Janet basis from it. In principle this leads to the same answer. However, the resulting system contains several fairly complicated equations for which a solution algorithm does not seem to exist. This is particularly obvious if B2

274 and C2 are substituted by their definition in terms of σ and ρ into the system (6.36). Lemma 6.7 shows clearly the group theoretical origin of the proceeding described in the above proof, it follows closely the discussion by Lie [109], part III, pages 364-370. The existence of the crucial relations (6.38) may be understood as follows. In addition to the six second order invariants the group (6.34) has eight independent third order invariants. Due to the fact that the twelve first derivatives of the second order invariants are also invariants of third order, four relations must exist between them. For example, by differentiation and some rearrangement the difference C2,x − Dy may be written as C2,x − Dy =


∆x ∆y 2 σxx ρxy − σxy ρxx − C2 + D ∆ ∆ ∆

∆y x with ∆ = σx ρy − σy ρx . Using ∆ ∆ = C2 − C1 and ∆ = B1 − B2 and applying some simplifications, the relation C2,x − Dy = (B1 + B2 )D − (C1 + C2 )C2 is obtained which is the first equation of (6.38). Due to the group structure they are obtained as integrability conditions during the Janet basis algorithms. Example 6.14

Consider equation 6.180

x2 (y − 1)y 00 − 2x2 y 02 − 2x(y − 1)y 0 − 2y(y − 1)2 = 0 2 , C = − 2 and of the collection by Kamke with A = 0, B = − y − x 1 2y(y − 1) 2 D = − . By Theorem 5.34 it belongs to symmetry class S8 . The x2 system (6.37) for a and b is 2 a + 2y(y − 1) b + 2 = 0, a + ab = 0, ax + a2 − x y x2 x2 2 b = 0. bx + ab = 0, by + b2 + y − 1 Its general solution is rational and may be written as y − 1 2x(y − 1) + 1 y y a= + , 2 x x(y − 1) x (y − 1) C1 + C2 + y y C2

x x2 2 + 2 1 y y b= − 2 y(y − 1) x(y − 1) x (y − 1) C1 + C2 + y y C2

Transformation to Canonical Form

275

where C1 and C2 are the integration constants. For the values C1 → ∞ and 1 C2 = 0 the special solution a = 0 and b = − is obtained. They yield y(y − 1) the Janet basis σxx +

2y(y − 1) 2 σ = 0, σ − 1 σy − x σ = 0, σyy + y2 σy = 0 x xy y(y − 1) x x2

for σ and an identical one for ρ. It is completely reducible with the Loewy decomposition  Lclm σx −

 (C2 + 2x)(y − 1) (C2 + x)x σ, σy − σ = 0. 2 C1 y + (C2 + x)x(y − 1) C1 y + (C2 + x)xy(y − 1)

Choosing C1 → ∞, C2 = 0 or C1 = C2 = 0 or C1 = 0, C2 → ∞ leads to the representation Lclm



   2 1 1 1 σx , σy , σx − σ, σy − 2 σ , σx − σ, σy − 2 σ =0 x x y −y y −y

 x(y − 1) x2 (y − 1) , is obtained. from which the fundamental system 1, y y Therefore possible transformation functions are u=

x2 (y − 1) v v x(y − 1) . , v= with inverse x = , y = y y u v − u2

The projective symmetry entails that any transformation σ ¯=

a1 σ + a2 ρ + a3 , a7 σ + a8 ρ + a9

ρ¯ =

a4 σ + a5 ρ + a6 a7 σ + a8 ρ + a9

generates the same equation 6.180 from the canonical form v 00 (u) = 0. For example, choosing ak = 1 for k = 1, 3, 5, . . . , 9 and a2 = a4 = 2 yields the transformation u≡σ ¯=

x(2x + 1)(y − 1) , y

v ≡ ρ¯ =

x(x + 2)(y − 1) y

and its reverse x≡φ=

v − 2u 3(v − 2u , y≡ψ= 2 . u − 2v u − 4uv − 6u + 4v 2 + 3v

If it is applied to equation 6.180 it generates −

(2u − v)3 (u − 2v)6 v 00 = 0, 3[(4v + 3)v + u2 − 2(2v + 3)u]3 (uv 0 − v)3

i.e., up to a lower order factor the canonical form v 00 = 0.

276 Example 6.15

As a second example consider equation 6.124 yy 00 − 3y 02 + 3yy 0 − y 2 = 0

of the collection by Kamke where A = 0, B = − y3 , C = 3 and D = −y. By Theorem 5.34 it belongs to symmetry class S82 . The system (6.37) is 3 ax + a2 + 3a + by + 2 = 0, ay + ab = 0, bx + ab = 0, by + b2 + b = 0. y Its rational solutions are a = −1, b = 0 or a = −2, b = 0 or a = 0, b = − y2 . Choosing the first alternative yields B2 = 0, C2 = 1 and leads to the Janet basis 3 σxx + yσy + σx = 0, σxy − σy = 0, σyy + σy = 0. y It is completely reducible and may be represented as    2  Lclm σx , σy , σx + σ, σy , σx − σ, σy + σ = 0. y  x There is an identical one for ρ. They yield the fundamental system 1, e−x , e 2 . y A possible choice for the transformation functions is u = e−x , v =

ex 1 1 √ . 2 with inverse x = log u , y = uv y

It is instructive to evaluate the same example after a transformation to a new y¯ 1 set of variables has been performed. Introducing x ¯ and y¯ by x = x ¯, y = x ¯ and renaming them x and y, the equation 6.124 becomes 3 6 3 3 3 1 3 3 1 y 00 − y 03 + 2 y 02 y − y 02 − 3 y 0 y 2 + 2 y 0 y − y 0 + 4 y 3 − 3 y 2 + 2 y = 0. x x x x x x x x x There is a third order equation for the x-dependence of w now with the Loewy decomposition Lclm

h

wx , wxx +

4 x

−

 3 2x − y  1 2y y 2 3x − 2y  i wx + − 2+ 3− 4− w =0 P xy x yP x x

where P ≡ x2 − xy + 13 y 2 . The first order factor yields the Janet basis σxxx +

6x2 − 9xy + 4y 2 6x4 − 12x3 y + 10x2 y 2 − 5xy 3 + y 4 σ − σx = 0, xx 3x3 − 3x2 y + xy 2 3x6 − 3x5 y + x4 y 2

σy −

x2 y −

1 4 3x σx,x xy 2 + 13 y 3

+

1 3 2 2 1 2 3 x − 3 x y + 3 xy σx x2 y − xy 2 + 31 y 3

= 0.

Transformation to Canonical Form

277

It is completely reducible with the representation  2 2 3 1 σ + 3x y − 3xy + y σ, Lclm σxx − x − 5 4 y x x −x y  x σ + 1 σ , σ , σ } = 0. σy − x − x x y y x−y In some applications there occur equations of the general form (5.34) with A = 0. This may be due to the fact that the transformation to canonical form is achieved by a so-called fiber-preserving transformation, i.e., the transformation function σ is independent of y. This specialization leads to a considerable simplification of the the procedure described in the above Theorem 6.14. The details are given in the subsequent corollary. Corollary 6.3 (Hsu and Kamran 1989) A second order ode of the form y 00 + B(x, y)y 02 + C(x, y)y 0 + D(x, y) = 0 is similar to v 00 (u) = 0 by a fiberpreserving point transformation if and only if its coefficients satisfy
Cy − 2Bx = 0, Cx + 12 C 2 − 2Dy − 2BD y = 0. The transformation function u = σ(x) and v = ρ(x, y) are determined by ρ = 0, ρxy − 12 Cρy − 12 σσxx ρ = 0, ρyy − Bρy = 0, ρxx − Dρy − σσxx x x x x σxx
− 1 σxx
2 + C + 1 C 2 − 2D − 2BD = 0. x y σx x 2 σx 2 Proof Applying a general fiber-preserving transformation u = σ(x) and v = ρ(x, y) to v 00 = 0 yields σx ρy y 00 + σx ρyy y 02 + (2σx ρxy − σxx ρy )y 0 + σx ρxx − σxx ρx = 0 from which the relations σxx σxx  ρyy ρxy ρx  ρxx = B, 2 − = C, − =D ρy ρy σx ρy ρx σx follow. Transforming this system of pde’s into a Janet basis in lexicographic term ordering with ρ > σ > B > C > D and y > x yields the above equations for σ and ρ. The two lowest equations in the Janet basis are the coefficient constraints. It should be noticed that the Riccati equation for σσxx does not depend on x y due to the second coefficient constraint. In this form the above corollary solves the equivalence problem for second order equations with a maximal number of fiber-preserving symmetries constructively, i.e., for all equations that occur Liouvillian solutions may be obtained by the algorithms described in Chapter 2. Canonical Form Transformation and Infinitesimal Generators. In his main publication on solving differential equations Lie determines at first the

278 symmetry generators for a given second order ode explicitly, and then applies them for determining its canonical form. The following theorem is based on Lie [113], Kapitel 18, pages 412-425 and Kapitel 23, pages 503-531. Theorem 6.15 (Lie 1891) Let ξi ∂x +ηi ∂y , i = 1, . . . , r, r = 1, 2 or 3, be the infinitesimal generators for a second order ode obeying canonical commutation relations. In terms of these generators, the transformation functions σ and ρ are determined by the following systems of equations where ∆ij = ξi ηj − ξj ηi . S12 : ξ1 σx + η1 σy = 0, ξ1 ρx + η1 ρy = 1. 2 S2,1 : σx =

2 S2,2 : σx +

ξ2 η1 ξ1 η2 , σy = − , ρx = − , ρy = . ∆12 ∆12 ∆12 ∆12

ξ1 η1 η2 ξ1 ξ2 η1 σ = 0, σy − σ = 0, ρx + ρ= , ρy − ρ=− . ∆12 ∆12 ∆12 ∆12 ∆12 ∆12 2 S3,1 : σ+ρ=

2 S3,2 : σ2 =

2 S3,3 : σ=−

∆13 ∆23 . , σρ = ∆12 ∆12

∆13 η1 ξ1 , ρx + ρ = 0, ρy − ρ = 0. ∆12 ∆12 ∆12

∆13 ∆23 , cρ = . ∆12 ∆12

2 S3,4 : σ=

∆13 + ∆23 ∆13 , ρ=− . ∆12 ∆12

Proof For a one-parameter symmetry the solution is given by Lemma 3.4 on page 130. For two-parameter symmetries the proof has been given in 2 Lemma 3.5 on page 132. For symmetry class S2,1 corresponding to group g26 , the system ξ1 σx + η1 σy = 1, ξ2 σx + η2 σy = 0, ξ1 ρx + η1 ρy = 0, ξ2 ρx + η2 ρy = 1 2 is obtained; and for symmetry class S2,2 corresponding to group g25 it is

ξ1 σx + η1 σy = 0, ξ2 σx + η2 σy = σ, ξ1 ρx + η1 ρy = 1, ξ2 ρx + η2 ρy = ρ. In both cases the above result follows by algebraic elimination of the first order derivatives. The transformstion functions are obtained in terms of path integrals. For the three-parameter groups, due to N = 3 in Lemma 2.12, two algebraic 2 relations for σ and ρ are obtained. For the symmetry S3,1 the systems for σ and ρ are ξ1 σx + η1 σy = 1, ξ2 σx + η2 σy = σ, ξ3 ρx + η3 ρy = σ 2 , ξ1 ρx + η1 ρy = 1, ξ2 ρx + η2 ρy = ρ, ξ3 ρx + η3 ρy = ρ2

Transformation to Canonical Form

279

leading to ∆12 σ 2 − ∆13 σ + ∆23 = 0, ∆12 ρ2 − ∆13 ρ + ∆23 = 0 2 with the solution as given above. For the symmetry class S3,2 the systems for σ and ρ are

ξ1 σx + η1 σy = 1, ξ2 σx + η2 σy = 2σ, ξ3 ρx + η3 ρy = σ 2 , ξ1 ρx + η1 ρy = 0, ξ2 ρx + η2 ρy = ρ, ξ3 ρx + η3 ρy = σρ2 . The two algebraic relations ∆12 σ 2 + 2∆23 σ = 0, ∆12 σρ − ∆13 ρ = 0 ∆23 2 13 yield σ = ∆ ∆12 and σ = ∆12 . The second function ρ is not determined by these relations. Therefore, the first two equations for ρ have to be solved for 2 its first derivatives with the above result. For symmetry class S3,3 the system for σ and ρ is ξ1 σx + η1 σy = 1, ξ2 σx + η2 σy = 0, ξ3 ρx + η3 ρy = σ, ξ1 ρx + η1 ρy = 0, ξ2 ρx + η2 ρy = ρ, ξ3 ρx + η3 ρy = cρ2 2 leading to ∆12 σ + ∆23 = 0 and c∆12 − ∆13 = 0. Finally, for S3,4 the system

ξ1 σx + η1 σy = 0, ξ2 σx + η2 σy = 0, ξ3 ρx + η3 ρy = σ, ξ1 ρx + η1 ρy = 1, ξ2 ρx + η2 ρy = 0, ξ3 ρx + η3 ρy = σ + ρ2 leads to ∆12 σ − ∆13 = 0 and ∆12 (σ + ρ) + ∆23 = 0 from which the above relations follow immediately. This result shows clearly the decreasing complexity of finding the canonical variables with increasing number of symmetries. For the full problem, however, it must be taken into account that symmetry generators have to be determined first.

6.3

Nonlinear Third Order Equations

For second order equations there is only a single equivalence class with a linear canonical representative which may be chosen as v 00 = 0. This equivalence class makes up the full symmetry class S82 . The situation is different for equations of order higher than two. There are infinitely many equivalence classes with linear canonical representatives that may be combined to form

280 several symmetry classes. As it has been shown in Chapter 2, there are special solution procedures for linear equations that do not exist for nonlinear ones. Therefore, it appears reasonable to consider the two cases separately. The subject of this Section 6.3 is third order equations that are genuinely nonlinear, i.e., they are not equivalent to a linear equation. These latter equations will be treated in the subsequent Section 6.4. Canonical Forms and Structure Invariance Groups. Similar to equations of order two, the tabulation of the groups of the plane including its invariants allows determining the possible forms of invariant equations. They are described in the subsequent theorem together with its structure invariance groups. Theorem 6.16 In canonical variables u and v ≡ v(u) the third order quasilinear equations with nontrivial symmetries have the following canonical forms and corresponding structure invariance groups of point transformations u = σ(x, y) and v = ρ(x, y); f , g and r are undetermined functions of the respective arguments, a, ai for all i and c are constants. One-parameter symmetry group S13 : v 000 + r(u, v 0 , v 00 ) = 0 allows the pseudogroup u = f (x), v = g(x) + cy. Two-parameter symmetry groups 3 S2,1 : v 000 + r(v 00 , v 0 ) = 0 allows u = a1 x + a2 y + a3 , v = a4 x + a5 y + a6 . 3 S2,2 : u2 v 000 + r(uv 00 , v 0 ) = 0 allows u = a1 x, v = a2 x + a3 y + a4 .


00 3 S2,3 : v 000 + v 0 r v 0 , u = 0 allows u = f (x), v = a1 y + a2 . v 3 S2,4 : v 000 + r(v 00 , u) = 0 allows

(x + a1 )a2 , 1 − (x + a1 )a2 a3

σ=

ρ=

a3 y + f (x). 1 − (x + a1 )a2 a4

Three-parameter symmetry groups (10)

3 : S3,1

Φ1

=

[(u − v)v 00 + 2v 0 (v 0 + 1)]2 , v 03

(u − v)2 v 000 + 6(u − v)(v 0 + 1)v 00 6(v 02 + 4v 0 + 1) + . 02 v0 v (10) (10)
ω Φ 1 , Φ2 = 0 allows (10)

Φ2

=

u=

a1 (x + a2 ) a1 (y + a2 ) , v= . 1 − a1 a3 (x + a2 ) 1 − a1 a3 (y + a2 )

Transformation to Canonical Form

281

3 S3,2 : v 000 v 5 + 3v 00 v 0 v 4 + r(v 00 v 3 ) = 0 allows

u=

a1 (x + a3 ) a2 y , v= . 1 + a4 (x + a3 ) 1 + a4 (x + a3 )


3 S3,3 (c): v 000 + v 0(c−3)/(c−1) r v 00 v 0(2−c)/(c−1) = 0 allows u = a1 x + a2 , v = a3 y + a4 . 0

0

3 S3,4 : v 000 e2v + r(v 00 ev ) = 0 allows u = a1 x + a2 , v = a3 x + a1 y + a4 . 00
3 S3,5 : v 000 + v 0 r v 0 = 0 allows u = a1 x + a2 , v = a3 y + a4 . v 3 S3,6 : v 000 + r(v 00 ) = 0 allows u = a1 x + a2 , v = a3 y + a4 x2 + a5 x + a6 . 3 S3,7 : v 000 − 2v 00 + v 0 + r(v 00 − 2v 0 + v) = 0 allows u = x + a1 ,

v = a2 y + a3 + a4 ex + a5 xex . 3 S3,8 : v 000 − v 00 + r(v 00 − v 0 ) = 0 allows u = x + a1 , v = a2 y + a3 ex + a4 x + a5 . 3 S3,9 : v 000 − (α + 1)v 00 + αv 0 + r[v 00 − (α + 1)v 0 + αv] = 0 allows u = x + a1 , v = a2 y + a3 + a4 ex + a5 eαx . 3 S3,10 : v 000 + v 002 r(v 0 ) = 0 allows u = a1 x + a2 y + a3 , v = a4 x + a5 y + a6 .

Four-parameter symmetry groups 3 S4,1 : v 000 v 0 + av 002 = 0 with a 6= 0, − 23 allows u = a1 x + a2 , v = a3 y + a4 . √ a1 x + a , v = a4 y . 3 S4,2 : v 000 v + 3v 00 v 0 + av 00 v 00 v = 0 allows u = 1 − 3 a2 x 1 − a2 x 3 S4,3 (c): v 000 + av 00(c−3)/(c−2) = 0 allows u = a1 x + a2 , v = a3 x + a4 y + a5 . 1

00

3 S4,4 : v 000 + ae− 2 v = 0 allows u = a1 x + a2 , v = a21 y + a3 x2 + a4 x + a5 .

Six-parameter symmetry group S63 : v 000 v 0 − 32 v 002 = 0 allows u = a1 + a2 x + a3 x2 , v = a4 + a5 y + a6 y 2 . The proof is similar to that for Theorem 6.4 and is not given. A few symmetry types are considered in the exercises. One- and Two-Parameter Symmetries. The symmetry class S13 and 3 3 the first two symmetry classes S2,1 and S2,2 described in Theorem 5.13 correspond to the same groups as the respective symmetry classes for second order equations. Its transformation to canonical form is therefore known by Theorems 6.8 and 6.9.

282 Example 6.16 The equation considered in Example 5.27 in symmetry (2,2) class S13 generates a type J1,2 Janet basis. Therefore case b) of Theorem 6.7 dy = 0 yields σ = y, and the path integral leads to applies. The equation dx ρ = − log x. Substituting these two functions the canonical form v 000 v 0 − 3v 002 − (u − 4)v 00 v 02 + (2u − 4)v 04 + v 03 = 0 is obtained. Example 6.17 The equation considered in Example 5.28 in symmetry class (2,2) 3 S2,1 generates a type J2,3 Janet basis. Therefore case b) of Theorem 6.8 applies. It yields the system σxx = 0, σxy − y1 σx = 0, σyy = 0 and an identical system for ρ with a solution basis {1, y, xy}. The choice σ = y and ρ = xy v , y = u and the canonical equation v 000 + 2v 00 v 0 + 1 = 0. leads to x = u Example 6.18 The equation considered in Example 5.29 in symmetry (2,2) 3 class S2,2 generates a type J2,3 Janet basis. The expressions (6.6) and (6.8) of Theorem 6.9 yield σ = y1 and ρ = x from which the canonical form u2 v 000 −

3 2 002 u v − 5uv 00 − v 03 − 4v 0 = 0 v0

is obtained. 3 3 The remaining symmetry classes S2,3 and S2,4 of this subsection are treated in the following two theorems. 3 Theorem 6.17 If a third order ode belongs to symmetry class S2,3 , the transformation functions σ and ρ may be determined as follows; Ci are constants. (2,2)

a) Janet basis type J2,2 : σ = f (x), f is an undetermined function of x, ρ obeys ρx = 0, ρyy − b1 ρy = 0. (6.39) (2,2)

b) Janet basis type J2,1 : σ is the same as in the preceding case, ρ obeys  a  1,x ρy + a1 ρx = 0, ρxx + 2 − b1 ρx = 0. (6.40) a1 (2,2)

c) Janet basis of type J2,5 : σ(x, y) = f (ϕ), ϕ(x, y) = C0 is the integral of dy = −a1 , (6.41) dx ρ obeys  a  1,y − c1 ρy = 0. (6.42) ρx = 0, ρyy + 2 a1

Transformation to Canonical Form

283

(2,2)

d) Janet basis type J2,4 : σ is the same as in the preceding case, ρ obeys ρy + b1 ρx = 0, ρxx +

h 2(a b ) i 1 1 x − c1 ρx = 0. a1 b1 + 1

(6.43)

Proof In all four cases the equations for ρ are obtained as part of the Janet basis expressing the transformation functions in terms of the actual coefficients. In case a) and b) σ obeys σy = 0; in case c) and d) it obeys σy − a11 σx = 0 from which the given expressions follow from Corollary 2.7 on page 78. Example 6.19 The equation considered in Example 5.30 in symmetry (2,2) 3 class S2,3 generates a type J2,4 Janet basis. The expressions (6.6) and (6.8) of Theorem 6.9 yield σ = y1 and ρ = x from which the canonical form v 000 + v 0

h v 00 3 v0

i +u+1 =0

is obtained. 3 Theorem 6.18 If a third order ode belongs to symmetry class S2,4 , three types of Janet bases may occur; Ci are constants. R R (2,2) a) Janet basis type J2,1 : σ = C1 exp ( S(x)dx)dx + C2 , S 6= 0 is a solution of

Sx − 12 S 2 + b1,x + 21 b21 − 2b2 = 0,

Sy = 0

(6.44)

or σ = C1 x + C2 if S = 0. R R R R ρ = C3 exp ( 12 (S + b1 )dx − a2 dy) exp ( a2 dy)dy + f (x) where f (x) is an undetermined function of its argument. R (2,2) b) Janet basis type J2,5 : σ = C1 exp ( R(y)dy) + C2 , R 6= 0 a solution of Ry − 21 R2 + c1,y + 21 c21 − 2c2 = 0, Rx = 0 (6.45) or σ = C1 y + C2 if R = 0. R R R R ρ = C3 exp ( 12 (R + c1 )dy − b1 dx) exp ( b1 dx)dx + g(y) where g(y) is an undetermined function of its argument. (2,2)

c) Janet basis type J2,4 : σ obeys the system σy +b1 σx = 0, σxx −Sσx = 0 where S is a solution of Sx − 21 S 2 + c1,x + 12 c21 − 2c2 = 0.

(6.46)

284 If σ is known, ∆ ≡ σx ρy − σy ρx is determined from   1 ∆x − 23 σσxx c ∆ = 0, + a b + 1 2,x 1 2 x   σ 1 1 1 ∆y − 32 σyy b + a b + b − b c ∆=0 + 1,x 2 1,y 2 1 1 2 2 2 y and, finally, Z ρ=

∆(φ(¯ x, y), y) dy σx (φ(¯ x, y), y) x ¯=σ(x,y)

(6.47)

where φ(¯ x, y) is defined as the inverse of x ¯ = σ(x, y). Proof

In case a) for σ and ρ the Janet basis   σy = 0, ρxy − 21 σσxx + b ρy = 0, ρyy − a2 ρy = 0, 1 x 2 σxxx σx − 32 σxx + (b1,x + 21 b21 − 2b2 )σx2 = 0

is obtained. Defining S ≡ σσxx , the last equation becomes the Riccati equax tion for S given above with the general solution S ≡R S(x, C R 0 ), C0 the integration constant. From the third equation ρ = C¯3 (x) exp ( a2 dy)dy + f (x) is obtained. Substituting the derivatives ρy and R ρxy into the second equation yields the linear first order equation C¯3,x + ( a2,x dy − 21 S − 12 b1 )C¯3 = 0 for C¯3 . Consequently, Z  Z  1 ¯ C3 = C3 exp 2 (S + b1 )dx − a2,x dydx with C3 constant. Substituting this value into the expression for ρ completes case a). In case b) the equations for σ and ρ are   σyy σx = 0, ρxx − b1 ρx = 0, ρxy − 21 σy + c1 ρx , 2 + (c1,y + 12 c21 − 2c2 )σy2 = 0. σyyy σy − 32 σyy

A similar calculation as in case a) leads to the above result. In case c) the equations for σ are
2 σy + b1 σx = 0, σxxx σx − 23 σxx + c1,x − 2c2 + 12 c21 σx2 = 0 is defined. The system from which the given representation follows if S ≡ σσxx x for ρ is     σxx + c 1 ρxy + b1 ρxx − 12 σσxx + 2a b + c ρ − b ρx = 0, 2,x 1 1 y 1 1 2 σx x   ρyy − b21 ρxx + b1 σσxx − a2,y + a2,x b1 + c1 + a2 b2 ρy x   +b21 σσxx + c + a b ρx = 0. 1 2 2 x

Transformation to Canonical Form

285

Defining ∆ as above and expressing the derivatives of ρ in terms of ∆, the above system for ∆ is obtained. If σ and ∆ are known, ρ may be determined from it applying part ii) of Corollary 2.7. Example 6.20 The equation considered in Example 5.31 in symmetry (2,2) 3 class S2,4 generates a type J2,5 Janet basis. Consequently, case b) applies. 1. R = 0 leads to the special solution σ = y. The corresponding value for ρ is x 000 002 00 From these transformation functions the canonical form v +uv +v +1 = 0 follows. Three-Parameter Symmetries. The first four symmetry classes in this 3 3 subsection are S3,1 , . . . , S3,4 . They correspond to the respective symmetry classes of second order equations, and it has been described in Theorems 6.10 to 6.13 how the transformation functions to canonical form are determined. These results may be applied to third order equations without any change. Example 6.21 The canonical form transformation of the ode in symmetry 3 class S3,1 considered in Example 5.32 is covered by Theorem 6.10, case a). Because b1 = 0, P and Q are determined from (6.15) with the result P = 2 . R is obtained from (6.16) with the result R = x 1 . ,Q= x− y y(x − y) (x − y)2 The system for σ is σx = 0, σyy + y2 σy = 0 with the non-constant solution 1 follows. With these transformation functions the σ = y1 from which ρ = x canonical form v 000 +

0 6v 00 (v 0 + 1) + 6v 2 (v 02 + 4v 0 + 1) u−v (u − v) h 00 i 02 (v (u − v) + 2v 0 (v 0 + 1))2 + v 2 1 =0 + (u − v) v 03

is obtained. Example 6.22 The canonical form transformation of the ode in symmetry 3 class S3,2 considered in Example 5.33 is covered by Theorem 6.11, case b). At first Ryy + Ry2 + y2 Ry = 0 yields the special solution Ry = − y1 . With 1 the system ρ − 1 ρ =, ρ − 1 ρ = is obtained with the solution Rx = − x x y x y ρ = xy. Finally, σx = σyy = 0 with the special solution σ = y leads to the canonical form v 000 v 5 + 3v 00 v 0 v 4 + 2v 00 v 2 + 1 = 0. Example 6.23 The canonical form transformation of the ode in symmetry 3 class S3,3 (c) considered in Example 5.34 is covered by Theorem 6.12. Because (2,2) y b1 = 2x 6= 0 of its type J3,6 Janet basis, the last case of a) applies. The 2 σ = 0 and ρ + x ρ = 0, ρ + x ρ = 0 systems for σ and ρ are σy = 0, σxx + x x y y x xx y x y 1 from which the transformation functions σ = x and ρ = x are obtained. They yield the canonical form v 000 v 03 + 1 = 0.

286 Example 6.24 The canonical form transformation of the ode in symmetry 3 class S3,4 considered in Example 5.35 is covered by Theorem 6.13, case a). Due to b1 = c1 = −1, the system for σ is σx = σyy = 0 with the special solution σ = y. The remaining equations ρx = σy , ρyy = 0 yield ρ = x and 0 the the canonical form v 000 + v 002 + e−2v = 0 follows. 3 3 The remaining six three-parameter symmetries S3,5 , . . . , S3,10 do not have a counterpart for second order equations and are treated next. For the sym3 metry class S3,9 (c) the parameter c has been determined in Theorem 5.14 in terms of Janet basis coefficients; i.e., the symmetry class may be considered as known. The proof of Theorem 6.19 to 6.24 is essentially identical in all cases, the given equations are immediately obtained as part of a Janet basis for the pde’s expressing the general determining system in terms of the respective Janet basis coefficients. 3 Theorem 6.19 If a third order ode belongs to symmetry class S3,5 , the transformation functions σ and ρ may be determined by the following systems of pde’s. (2,2)

a) Janet basis type J3,5 : σy = 0, σxx − a2 σx = 0, ρx = 0, ρyy − d1 ρy = 0.

(6.48)

(2,2)

b) Janet basis type J3,4 : σy = 0, σxx − a3 σx = 0, ρy + c1 ρx = 0, ρxx − (2c3 − 2a3 − d1 )ρx = 0. (6.49) (2,2)

c) Janet basis type J3,7 : There holds always c1 6= 0. σy −

1 σx = 0 = 0, σxx −a3 σx = 0, ρx = 0, ρyy −(2a2 −2c2 −d1 )ρy = 0. c1 (6.50) (2,2)

d) Janet basis type J3,6 : If b1 = 0, the system is σx = 0, σyy − c2 σy = 0, ρy + a1 ρx = 0, ρxx − d1 ρx = 0

(6.51)

and if b1 6= 0 σy − 1 σx = 0, σxx − b3 σx = 0, b1 b1

(6.52)

ρy + a1 ρx = 0, ρxx + (2a1 b3 − 2a3 b1 − d1 )ρx = 0. Example 6.25 The Janet basis coefficients given in Example 5.36 lead to 2 ρ = 0. A functionally the systems σx = 0, σyy + y2 σy = 0 and ρy = 0, ρxx + x x

Transformation to Canonical Form

287

1 . It yields the canonical form independent set of solutions is σ = y1 , ρ = x v 000 v 0 + v 002 − 2v 02 = 0. 3 Theorem 6.20 If a second order ode belongs to symmetry class S3,6 , the transformation functions σ and ρ are determined by the following systems of pde’s. (2,2)

a) Janet basis type J3,4 : σy = 0, σxx − a3 σx = 0, ρxy − c3 ρy = 0, ρyy − c2 ρy = 0, ρxxx − 3a3 ρxx − d3 ρy − (a3,x − 2a23 )ρx = 0. (2,2)

b) Janet basis type J3,7 : σx = 0, σyy − c2 σy = 0, ρxx − a3 ρx = 0, ρxy − 12 (c2 + d1 )ρx = 0, ρyyy − 3c2 ρyy + (d1,y + 32 c22 + 21 d21 − 2d3 )ρy − d2 ρx = 0. (2,2)

c) Janet basis type J3,6 : There holds always a1 6= 0. σy + a1 σx = 0, σxx + a1 b3 σx = 0, ρxy + a1 ρxx − (a1 b3 − a3 b1 + b2 )ρy − a3 ρx = 0, ρyy − a21 ρxx + (a21 b3 + a3 − c2 )ρy + (a1 a3 − a2 )ρx = 0, a3 b + d3
ρ ρxxx + 3a1 b3 ρxx − b3,x + 3a1 b23 + b2 b3 + 2 a a1 y 1 3 −(2a3 b3 + d3 )ρx = 0. 3 An example for a canonical form transformation in symmetry class S3,6 is given following Theorem 6.23 in conjunction with examples for symmetry 3 3 3 , S3,8 and S3,9 (c) which are treated in the next three theorems. classes S3,7 3 Theorem 6.21 If a second order ode belongs to symmetry class S3,7 , the transformation functions σ and ρ are determined by the following systems of pde’s. (2,2)

a) Janet basis type J3,4 : σy = 0,

σx = c3 − 12 (a3 + d1 ),

ρxy − c3 ρy = 0, ρyy − c2 ρy = 0,

ρxxx + (d1 − 2a3 − 2c3 )ρxx − d3 ρy −(a3,x − 54 a23 − 41 d21 + d1 c3 + 12 a3 d1 − c23 − a3 c3 )ρx = 0.

288 (2,2)

b) Janet basis type J3,7 : σx = 0, σy = a2 − 12 (c1 + d1 ), ρxx − a3 ρx = 0, ρxy − a2 ρx = 0, ρyyy − (2a2 + 2c2 − d1 )ρyy + (d1,y + a22 + a2 c2 − a2 d1 + 43 c22 − 12 c2 d1 + 43 d21 − 2d3 )ρy − d2 ρx = 0. (2,2)

c) Janet basis type J3,6 : There holds always a21 b3 − a1 d1 + 2a3 6= 0 and a1 6= 0. σx = a1 b3 − 2a3 b1 − d1 , σy = a1 d1 − a21 b3 − 2a3 , ρxy + a1 ρxx − (a1 b3 − a3 b1 + b2 )ρy − a3 ρx = 0, ρyy − a21 ρxx + (a21 b3 + a3 − c2 )ρy + (a1 a3 − a2 )ρx = 0, ρxxx +

P1 (x, y) P2 (x, y) P3 (x, y) ρxx + ρy + ρx = 0 Q(x, y) Q(x, y) 4Q(x, y)

where P1 (x, y) ≡ 2a31 b23 − a21 b3 d1 + 2a1 a3 b3 − a1 d21 + 4a23 b1 + 4a3 d1 ,

(6.53)

P2 (x, y) ≡ −b3,x a21 b3 + b3,x a1 d1 − 2b3,x a3 − 2a31 b33 − a21 b2 b23 +a21 b23 d1 − 4a1 a3 b23 + a1 b2 b3 d1 + a1 b3 d21 − a1 b3 d3

(6.54)

+2a3 b1 d3 − 2a3 b2 b3 − 2a3 b3 d1 + d1 d3 , P3 (x, y) ≡ a41 b33 − 3a31 b23 d1 − 2a21 a3 b23 + 3a21 b3 d21 − 4a21 b3 d3 −4a1 a3 b3 d1 − a1 d31 + 4a1 d1 d3 + 8a33 b21 +12a23 b1 d1 − 4a23 b3 + 6a3 d21 − 8a3 d3 , Q(x, y) ≡ a21 b3 − a1 d1 + 2a3 .

(6.55)

3 Theorem 6.22 If a second order ode belongs to symmetry class S3,8 , the transformation functions σ and ρ are determined by the following systems of pde’s. (2,2)

a) Janet basis type J3,4 : σx = 2c3 − a3 − d1 , σy = 0, ρxy − c3 ρy = 0, ρyy − c2 ρy = 0, ρxxx + (d1 − 2a3 − 2c3 )ρxx − d3 ρy −(a3,x − a23 + a3 d1 − 2a3 c3 )ρx = 0.

Transformation to Canonical Form

289

(2,2)

b) Janet basis type J3,7 : σx = 0, σy = 2a2 − c2 − d1 , ρxx − a3 ρx = 0, ρxy − a2 ρx = 0, ρyyy − (2a2 + 2c2 − d1 )ρyy + (d1,y − 2a22 + 4a2 c2 +2a2 d1 − 2c2 d1 − 2d3 )ρy − d2 ρx = 0. (2,2)

c) Janet basis type J3,6 : There holds always a1 6= 0. σx = a1 b3 − 2a3 b1 − d1 , σy = a1 d1 − a21 b3 − 2a3 , ρxy + a1 ρxx − (a1 b3 − a3 b1 + b2 )ρy − a3 ρx = 0, ρyy − a21 ρxx + (a21 b3 + a3 − c2 )ρy + (a1 a3 − a2 )ρx = 0, a3 )ρ ρxxx + (2a1 b3 + d1 − 2 a xx 1 d3 )ρ − 2a b ρ = 0. −(b3,x + b3 d1 + 2a1 b23 + b2 b3 + a y 3 3 x 1 3 Theorem 6.23 If a second order ode belongs to symmetry class S3,9 (c) the transformation functions σ and ρ are determined by the following systems of pde’s. (2,2)

a) Janet basis type J3,4 : σx =

1 (2c3 − a3 − d1 ), σy = 0, ρxy − c3 ρy = 0, ρyy − c2 ρy = 0, c+1

ρxxx + (d1 − 2a3 − 2c3 )ρxx − d3 ρy + [a3,x − a23 −

c (a2 + d21 ) (c + 1)2 3

2 4c c (c − d )]ρ = 0. − c + 12 a3 (2c3 − d1 ) − 3 3 1 x (c + 1) (c + 1)2 (2,2)

b) Janet basis type J3,7 : 1 (2a2 − c2 − d1 ), ρxx − a3 ρx = 0, ρxy − a2 ρx = 0, c+1 h c ρyyy + (d1 − 2a2 − 2c2 )ρyy + c2,y − c22 − (c2 + d21 ) (c + 1)2 2 i 2 4c a (a − d ) ρ = 0. − c + 12 c2 (2a2 − d1 ) − 2 2 1 x (c + 1) (c + 1)2 σx = 0, σy =

(2,2)

c) Janet basis type J3,6 : There holds always a1 6= 0. σx =

1 a3
1 a1 b3 − d1 + 2 (a1 d1 − a21 b3 − 2a3 ), , σy = c+1 a1 c+1

290 ρxy + a1 ρxx − (a1 b3 − a3 b1 + b2 )ρy − a3 ρx = 0, ρyy − a21 ρxx + (a21 b3 + a3 − c2 )ρy + (a1 a3 − a2 )ρx = 0, a3
ρ − b + 2a b2 + b b + b d + d3
ρ ρxxx + 2a1 b3 + d1 − 2 a xx 3,x 1 3 2 3 3 1 a1 y 1 h 2 c − d3 + 2 c + 12 a3 b3 − (a1 b3 − d1 )2 (c + 1) (c + 1)2 i − 4c 2 a32 (a3 − a1 d1 ) ρx = 0. (c + 1) a1 (2,2)

Example 6.26 The type J3,4 Janet bases for the ode’s that have been considered in Example 5.37 are dealt with in case a) of Theorems 6.20 to 2 σ = 0 are obtained 6.23. In the first case, for σ the equations σy = 0, σxx − x x  1 with the fundamental system 1, x . For the remaining three cases σ obeys 1 . For all four cases the σy = 0, σx = 12 with the special solution σ = − x x systems for ρ comprise the equations ρxy = 0 and ρyy = 0. In addition there is a third order equation     6 ρ + 6 ρ = 0, ρ 2 6 + 4 + 3 ρ = 0, 6 ρxxx + x x xx xxx + x + 2 ρxx + 2 x 2 3 4 x x x x x     6 + 1 ρ − 2 − 2 ρ =0 ρxxx + x xx x x3 x x2 or

2 4 8 6 ρxx − − 3 + 4 ρx = 0 2 2 x x x x x It is not necessary to obtain a fundamental system for ρ, rather it is sufficient to find a single solution such that σ and ρ are functionally independent. It 1 have this property and transform turns out that ρ = y together with σ = − x the ode’s of Example 5.37 into a canonical form. In all four cases it has the general structure dz z+z 2 +1 = 0 where z is defined by z ≡ v 00 , z ≡ v 00 −2v 0 +v, dx z ≡ v 00 − v 0 or z ≡ v 00 − 4v 0 + 3v. ρxxx +

6

+

3 Theorem 6.24 If a second order ode belongs to symmetry class S3,10 the transformation function σ is determined by the system of pde’s

σxx − b3 σy − (b2 − c3 )σx = 0, σxy − b2 σy − a3 σx = 0, σyy − (a3 + c2 )σy − a2 σx = 0.

(6.56)

The function ρ satisfies an identical system. Example 6.27 The Janet basis coefficients given in Example 5.38 lead 2 σ = 0, σ = 0, σ = 0 for σ and an identical one to the system σxx + x x xy yy  1 for ρ with the solution basis 1, x , y . According to the preceding theorem a 1 , ρ = y. It yields the proper choice for the transformation functions is σ = x 000 0 002 02 canonical form v v + v (v + 1) = 0.

Transformation to Canonical Form

291

Four-Parameter Symmetries. They occur for the first time for equations 3 of order three. For the symmetry class S4,3 (c) the parameter c has been determined in Theorem 5.14 in terms of Janet basis coefficients; i.e., the symmetry class may be considered as known. 3 Theorem 6.25 If a second order ode belongs to symmetry class S4,1 the transformation functions σ and ρ are determined by the following systems of pde’s. (2,2)

a) Janet basis type J4,10 : σx = 0, σyy − d1 σy = 0, ρy = 0, ρxx − c2 ρx = 0.

(6.57)

(2,2)

b) Janet basis type J4,9 : σy + b1 σx = 0, σxx + (b1 d2 − c2 )σx , ρy = 0, ρxx − c2 ρx = 0. (6.58) (2,2)

c) Janet basis type J4,14 : Two cases have to be distinguished. If a1 = 0 σy −

1 σx = 0, σxx − c2 σx = 0, ρx = 0, ρyy − (2b3 − c1 )ρy = 0 (6.59) b1

or an identical system with σ and ρ exchanged; and if a1 6= 0
√ σy − √ 1 σx = 0, σxx + b4 + a1 e2 − d2 −a1 σx = 0, −a1
√ ρy + √ 1 ρx = 0, ρxx + b4 + a1 e2 + d2 −a1 ρx = 0. −a1

(6.60)

Proof The given equations are immediately obtained as part of a Janet basis for the pde’s expressing the general determining system in terms of the respective Janet basis coefficients. In case c) for a1 = 0 it cannot be decided a priori which alternative applies. If u = σ and v = ρ, as obtained by the above system, does not yield the canonical form, the proper transformation is u = ρ and v = σ. The algebraic function that may occur for a1 6= 0 does not prevent the solution because only integrations are involved. (2,2)

Example 6.28 The coefficients of the type J4,14 Janet basis in Examy ple 5.39 single out the first alternative of case c). The system σy − x σx = 0, σxx = 0 yields σ = xy, and the system ρx = 0, ρyy + y2 ρy = 0 yields ρ = y1 . Substitution into the equation of Example 5.39 yields the canonical form v 000 v 0 − 21 v 002 = 0. Exchanging σ and ρ, the canonical form v 000 v 0 − 52 v 002 = 0 is obtained.

292 3 Theorem 6.26 If a second order belongs to symmetry class S4,2 the transformation functions σ and ρ are determined by the following systems of pde’s. (2,2)

a) Janet basis type J4,12 : Rx + 12 d2 = 0, Ryy + Ry2 − d1 Ry − 41 c1 d2 = 0, ρx + Rx ρ = 0, ρy + Ry ρ = 0, σx = 0, σyy − (2Ry + d1 )σy = 0. (6.61) (2,2)

b) Janet basis type J4,9 : Ry + b1 Rx + 21 b21 d2 + a4 − b1 c2 − 12 c1 = 0, Rxx + Rx2 + (b1 d2 − c2 )Rx + 14 d2 (b21 d2 + 2a4 − 2b1 c2 − c1 ) = 0, ρx + Rx ρ = 0, ρy + Ry ρ = 0, σy + b1 σx = 0, σxx + (2Rx + b1 d2 − c2 )σx = 0. (6.62) Proof In case a) the system of pde’s expressing the transformation functions in terms of the Janet basis coefficients comprises the following equations containing σ and ρ. ρ σx = 0, σyy − (2 ρy + d1 )σy = 0, ρx + 12 d2 ρ = 0, ρyy ρ − 2ρ2y − d1 ρy ρ + 41 c1 d2 = 0. Introducing the new function R by log ρ = −R leads to the equations given above. In case b) the equations obtained in the first place are ρ σy + b1 σx = 0, σxx − (2 ρx − b1 d2 c2 )σx = 0, ρy + b1 ρx + (b1 c2 + 12 c1 − a4 − 21 b21 d2 )ρ = 0, ρxx ρ − 2ρ2x + (b1 d2 − c2 )ρx ρ −( 12 a4 d2 + 41 a4 d2 + 14 b21 b22 − 12 b1 c2 d2 − 14 c1 d2 )ρ2 = 0. The same substitution log ρ = −R yields the equations for case b). 3 Theorem 6.27 If a second order ode belongs to symmetry class S4,3 (c) the transformation functions σ and ρ are determined by the following systems of pde’s. (2,2)

a) Janet basis type J4,9 : σy = 0, σxx − c2 σx = 0, ρxx −

d2 ρ − c ρ = 0, 2 x b2 + 2 y

ρxy + (b2 c2 − b4 )ρy = 0, ρyy − b3 ρy = 0.

(6.63)

Transformation to Canonical Form

293

(2,2)

b) Janet basis type J4,14 : 1 σ = 0, σ + (b + 1)(a + d − 2 a4 − c1 ) + b c σ = 0, σy − b2a+ x xx 2 3 1 2 2 x a2 a2 2 (6.64) 1 2a (b + 1)(a + d ) − (2b + 1)(2a + c ) + 2a b c ρ ρxx + 2 2 3 1 2 4 1 2 2 2 y 2b2 + 1   1 (b2 + 1)(2b4 + d1 − a3 ) + a2 b2 (b3 + d2 + e1 ) ρx = 0, (b2 + 1)(2b2 + 1) 1 ρ + 1 2a (b + 1) − a (a + 2b + c + d )ρ ρxy − b1a+ xx 4 2 2 3 4 2 1 y 2a2 2 1 (b + 1)(a + c + d )ρ = 0, + 2a 3 2 1 x 2 2 h i a2 e2 ρyy + 12 b2 (b3 + d2 + e1 ) − b3 − ρ + e2 ρ = 0. (b1 + 1)(2b2 + 1) y 2b2 + 1 x (6.65) Proof The given equations are immediately obtained as part of a Janet basis for the pde’s expressing the general determining system in terms of the respective Janet basis coefficients. In case a) the constraint c 6= −2 assures that the denominator b2 + 2 6= 0, see Theorem 5.13. (2,2)

The type J4,9

Janet basis in Example 5.40 singles out 2 case a). The system σy = 0, σxx + x σx = 0 yields the non-constant solution 1 . The system σ=x Example 6.29

ρxx −

2y 2 1 2 ρy + x ρx = 0, ρxy − x ρy = 0, ρyy = 0 x

yields the independent solution ρ = xy. Substitution into the equation of Example 5.40 leads to the canonical form v 000 + v 003 = 0. 3 Theorem 6.28 If a second order ode belongs to symmetry class S4,4 the transformation functions σ and ρ are determined by the following systems of pde’s. (2,2)

a) Janet basis type J4,9 : There holds always d2 6= 0. σy = 0, σxx − c2 σx = 0, ρy =

2σ 2 − x, d2


ρxxx − 3c2 ρxx + (2c22 − c4 )ρx = 2 c2 − d4 σx2 . d2

(6.66)

(2,2)

b) Janet basis type J4,14 : There holds always e2 6= 0. Two cases have to be distinguished. If b1 = 0 the system is σx = 0, σyy − (b3 + d2 )σy = 0, ρx = − e12 σy2 , ρyyy − 3(b3 + d2 )ρyy − b3,y + 12 e4

−2b23 − 3b3 d2 + b4 e2 − 3d22 ρy = 2d2 + ee32 σy2

(6.67)

294 and if b1 6= 0
σy − 3 σx = 0, σxx − 49 b21 e2 + 3 a4 + 3c1 σx = 0, 2b1 b1 2b1 (6.68)
3 ρy − 3 e2 ρx = 3 σx2 , 2b1 2b1
ρxxx − 1 9a4 + 43 b31 e2 + 92 c1 ρxx b1 64 6 2 + 12 32 c1,x b1 + 18a24 + 8a4 b31 e2 + 63 4 a4 c1 + 81 b1 e2 b1
+ 83 b31 c1 e2 + 32 b31 c3 + 2b21 c1 d2 − b21 c4 + 23 b1 b4 c1 + 92 c21 ρx =

σx2 − b41 e2

27 8 c1,x b1

+

243 16 a4 c1

+ 3b31 c1 e2 − 32 b31 c3 − 92 b21 c1 d2 −

27 8 b1 b4 c1




.

(6.69) Proof The given equations are immediately obtained as part of a Janet basis for the pde’s expressing the general determining system in terms of the respective Janet basis coefficients. Six-Parameter Symmetry. The only symmetry class is S63 with canonical form v 000 v 0 − 32 v 002 = 0. It is treated along similar lines as the projective symmetry S82 of second order equations. At first the invariants are determined from which the coefficients of an equation with this symmetry may be constructed. Lemma 6.8 The differential invariants up to order three for the group S63 generated by the vector fields {∂σ , ∂ρ , σ∂σ , ρ∂ρ , σ 2 ∂σ , ρ2 ∂ρ } and the transformation u = σ(x, y), v = ρ(x, y) may be described as follows. a) For σx 6= 0 and ρx 6= 0 there are two first order invariants, four second order invariants σy ρy I1 = , I2 = I1,x , I3 = I1,y , K1 = , K2 = K1,x , K3 = K1,y σx ρx and eight third order invariants     σxx 2 , I = σyyy − 3 σyy 2 , I = I , I = I , 3 I4 = σσxxx − 5 6 2,x 7 2,y 2 σx σy 2 σy x     ρyyy ρyy 2 ρ ρxx 2 3 3 K = K4 = ρxxx − , − , K6 = K2,x , K7 = K2,y . 5 2 ρx ρy 2 ρy x b) For σx = 0, ρx 6= 0 there is no invariant involving derivatives of σ up to order two and one invariant   σ σyy 2 3 J1 = σyyy − 2 σy y of order three with J1,x = 0. The invariants K1 , . . . , K7 involving ρ are the same as in the preceding case.

Transformation to Canonical Form

295

c) For σx 6= 0, ρx = 0 the invariants I1 , . . . , I7 involving σ are the same as in case a). There is no invariant involving derivatives of ρ up to order two and one invariant 2  ρ 3 ρyy − L1 = ρyyy ρy 2 y of order three with L1,x = 0. Proof

The prolongations of the group generators up to order three are

∂σ , σx ∂σx + σy ∂σy + σxx ∂σxx + σxy ∂σxy + σyy ∂σyy +σxxx ∂σxxx + σxxy ∂σxxy + σxyy ∂σxyy + σyyy ∂σyyy , σx2 ∂σxx + σx σy ∂σxy + σy2 ∂σyy + 3σx σxx ∂σxxx +(2σx σxy + σxx σy )∂σxxy + (2σxy σy + σx σyy )∂σxyy + 3σy σyy ∂σyyy and the same set with σ replaced by ρ. Due to the occurrence ∂σ and ∂ρ , any invariant is independent of σ and ρ. There are two equations involving the four first order derivatives. Consequently, there are two first order invariants. Up to order two there are twelve variables involved leading to six invariants altogether. Up to order three there are twenty variables involved leading to fourteen invariants up to this order. If the respective numbers of lower order invariants are taken into account the above figures are obtained. The functional independence is guaranteed due to the fact that each invariant depends on a different set of variables. The actual determination of the invariants is achieved by the methods described in Section 5 of Chapter 1. If σx = 0 and ρx 6= 0, the three prolongations involving derivatives of σ are ∂σ , σy ∂σy + σyy ∂σyy + σyyy ∂σyyy , σy ∂σyy + 3σyy ∂σyyy with the obvious solution J1 . If σx 6= 0 and ρx = 0, the same prolongations involving derivatives of ρ are obtained with the solution L1 . Lemma 6.9 The coefficients of an equation with S63 symmetry may be expressed in terms of the invariants introduced in the above lemma. a) For equation (5.40), σx 6= 0 and ρx 6= 0, the expressions for the coefficients are: A0 = I1 K1 , A1 = I1 + K1 , A2 = 3

I3 K12 − I12 K3 I1 − K1 ,

K12 I2 − I12 K2 + 2(K1 I3 − I1 K3 ) , I1 − K1 I − K3 + 2(K1 I2 − I1 K2 ) − K2 , A4 = 3 3 , A5 = 3 II2 − I1 − K1 K1 1 − K5 , B1 = I12 K12 II5 − K1 1 B2 = A0 (A0 + A21 )B6 − 2A0 A1 (2A0 + A1 )B7 , A3 = 3

296 B3 = A1 (3A0 + 12 A21 )B6 − (A20 + 7A0 A21 + A41 )B7 , B4 = 2(A0 + A21 )(B6 − 2A1 B7 ), B5 = 25 A1 B6 + (A0 − 4A21 )B7 , B6 =

3(I6 − K6 ) + 2(K1 I4 − I1 K4 ) + I1 I4 − K1 K4 K4 . , B7 = II4 − I1 − K1 1 − K1

b) For equation (5.41), σx = 0 and ρx 6= 0 the expressions for the coefficients are A0 = K1 , A2 = −3K3 , A3 = 3K2 , B1 = K12 (J1 − K5 ), B2 = 2J1 K1 − 3K13 K4 − 3K12 K6 + 23 K1 K22 − K1 K5 −3K1 K7 + 3K2 K3 +

3 2

K32 K1 ,

B3 = J1 − 6K12 K4 − 6K1 K6 + 3K22 − 3K7 , B4 = −4K1 K4 − 3K6 , B5 = −K4 . c) For equation (5.41), σx 6= 0 and ρx = 0 the expressions for the coefficients are A0 = I1 , A2 = −3I3 , A3 = −3I2 , B1 = I12 (L1 − I5 ), B2 = 2L1 I1 − 3I13 I4 − 3I12 I6 + 23 I1 I22 − I1 I5 −3I1 I7 + 3I2 I3 +

2 3 I3 2 I1 ,

B3 = L1 − 6I12 I4 − 6I1 I6 + 3I22 − 3I7 , B4 = −4I1 I4 − 3I6 , B5 = −I4 . The proof is essentially the same as for Lemma 6.9 except that the canonical form now is v 000 v 0 − 23 v 002 = 0. Therefore it is omitted. Theorem 6.29 The transformation of an equation in symmetry class S63 to canonical form may be determined as follows. a) If it has the form (5.40) the transformation functions σ and ρ are determined by the following system of pde’s. Sx − 12 S 2 − I4 = 0, Sy − 21 I1 S 2 − I1,x S − I1 I4 − I1,xx = 0, σy − I1 σx = 0, σxx − Sσx = 0,

(6.70)

Rx − 21 R2 − K4 = 0, Ry − 12 K1 R2 − K1,x R − K1 K4 − K1,xx = 0, ρy − K1 ρx = 0, ρxx − Rρx = 0. (6.71) The required invariants may be expressed in terms of the coefficients of (5.40) as follows. I1 = 21 (A1 + Z), I4 = 12 B6 − 21 (3A1 − Z)B7 − 32 ZZxx .

Transformation to Canonical Form

297

0 K p1 and K4 are obtained from the corresponding I s by replacing Z ≡ 2 A1 − 4A0 with −Z. Z is contained in the same field as the transformation functions.

b) If it has the form (5.41), two cases have to be distinguished. Either the transformation function σ is determined by the system Sx = 0, Sy − 12 S 2 − J1 = 0, σx = 0, σyy − Sσy = 0

(6.72)

where J1 = B3 − 6A20 B5 + 6A0 A0,xx − 3A20,x + 3A0,xy , and the transformation function ρ is obtained from system (6.71) where now K1 = A0 , K4 = −B5 . Or σ is determined by the system (6.70) and ρ is determined by the system Rx = 0, Ry − 12 R2 − J1 = 0, ρx = 0, ρyy − Rρy = 0

(6.73)

with J1 as above. is essentially the Proof Case a). The first equation of (6.70) for S ≡ σσxx x definition of I4 in Lemma 6.8. The second equation involving the y-derivative of S is obtained in the same way as in Theorem 6.33. The equations for σ are essentially the definitions of I1 and S. Similar arguments lead to the system ρ (6.71) for R ≡ ρxx . x σ in (6.72) is essentially the definition of Case b). The equation for S ≡ σyy y J1 in Lemma 6.8. In either case, the expressions for the invariants are obtained p by elimination from the relations given in Lemma 6.10. Writing Z as Z = A21 − 4A0 = ∆2 , ∆ = σ ρ − σ ρ , shows that it is contained in the field determined by x y y x σx2 ρ2x the transformation functions. Example 6.30 An equation in symmetry class S63 , case a), has been considered in Example 5.41 of the preceding chapter. From the coefficients x given there, the invariants I1 = x y , K1 = − y , I4 = K4 = 0 are obtained. x S2 − 1 S = 0 Consequently, the equations for S are Sx − 12 S 2 = 0, Sy − 2y y 2y with the general solution S = − xy + C . Choosing C → ∞ yields S = 0 and x R2 + 1 R = 0 with finally σ = xy. The system for R is Rx − 12 R2 = 0, Ry + 2y y 2 and the general solution R = − Cy2+ x . Choosing now C = 0 yields R = − x y 2 ρy + x y ρx = 0, ρxx + x ρx = 0 with the non-constant solution ρ = x . Example 6.31 An equation in symmetry class S63 , case b), has been considered in Example 5.42 of the preceding chapter. From the coefficients given

298 there, the invariants J1 = 0, K1 =

x 1 x2 − 4x + 6 , K4 = − (x − 1)y 2 (x − 1)2

are obtained. Consequently, the equations for S and σ are the same as in Example 6.39, the special solution σ = y1 is chosen again. R is determined by the system 2 Rx − 12 R2 + 21 x − 4x +2 6 = 0, (x − 1) (x − 2)(x2 − 2x + 2) 1 x Ry − 12 R2 + R + 21 = 0. 2 (x − 1)y (x − 1) y (x − 1)3 y 2 −2 It has the two rational solutions x − 2x + 2 and − x x − 1 . Choosing the first x(x − 1) alternative, the system

x x2 − 2x + 2 ρx = 0, ρxx − ρx = 0 xy − y x2 − x y for ρ is obtained with the non-constant solution ρ = x ex . ρy −

6.4

Linearizable Third Order Equations

The distinctive feature of the equations covered by this section is the fact that they allow canonical forms that are linear. Consequently, the solution procedures described in Chapter 2 for linear differential equations apply as soon as the transformation to canonical form has been found. At first the canonical forms and its structure invariance groups are described, after that the transformation to these canonical forms for the various symmetry classes is considered. Canonical Forms and Structure Invariance Groups. The subsequent theorem describes canoncial forms and structure invariance groups for the four symmetry types that occur in this section. Theorem 6.30 In canonical variables u and v ≡ v(u) the third order equations that are equivalent to a linear one and allow nontrivial symmetries have the following canonical forms and corresponding structure invariance groups of point transformations u = σ(x, y) and v = ρ(x, y); ai for all i are constant. Four-parameter symmetry group 3 1 x + a2 S4,5 : v 000 + P (u)v 0 + Q(u)v = 0 allows u = a a3 x + a4 ,

v=

y . (a3 x + a4 )2

Transformation to Canonical Form

299

Five-parameter symmetry groups 3 S5,1 : v 000 − v 00 = 0 allows u = x + a1 , v = a2 y + a3 x + a4 ex + a5 . 3 S5,2 (a): v 000 − (a + 1)v 00 + av 0 = 0 allows u = x + a1 , v = a2 y + a3 ex + a4 aa1 eax + a5 . In addition five transformations u = αx, v = eβu v with 1 , β = 0; α = β = −1; α = β = 1 ; α = 1 , β = 1 and α = a a a−1 1 ,β= a . α= a− 1 a−1

Seven-parameter symmetry groups 2

2 1 + a2 x , v = a4 + a5 x + a6 x + a7 y . S73 : v 000 = 0 allows u = ax + a3 x + a3 3 Proof The structure invariance group for the symmetry class S4,5 has 3 3 been given in Theorem 4.3. For the symmetry classes S5,1 and S7 they are identical to the respective symmetry groups as given in Theorem 5.16 because there are no parameters or undetermined functions involved.

Four-Parameter Symmetry. There is only a single symmetry class, the generic linear homogeneous third order ode belongs to it. The subsequent theorem describes how to transform a general equation in this symmetry class to a linear equation in rational normal form. 3 Theorem 6.31 If a third order ode belongs to the symmetry class S4,5 three types of Janet bases may occur. The transformation functions σ and ρ to rational normal form v 000 +P (u)v 0 +Q(u)v = 0 are determined as follows; f , g and h are undetermined functions of its arguments; A ≡ P (σ) and B ≡ Q(σ). (2,2)

a) Janet basis of type J4,2 : σ = f (x), the system for ρ is σ 1  xx + c1 ρy = 0, ρyy − b2 ρy = 0, σx 3 σxx ρxxx − 3 ρxx + c2 ρy σx h 3  σ 2 1 i 1 1 3 xx − + c1,x + c21 − c3 + σx2 A ρx + σx3 Bρ = 0. 2 σx 2 6 2 2 ρxy −

(6.74)

A and B are determined through σ   σ 2 1 xx xx 2 − +c1,x + c21 − c3 + σx2 A = 0, σx x σx 3 c1,xx + 2c1 c1,x − 3c3,x − 6b2 c2 + 94 c31 − 2c1 c3 + 6c4 + 3σx2 Ax − 6σx3 B = 0. (6.75)

300 (2,2)

b) Janet basis of type J4,19 : σ = g(y), the system for ρ is σ  yy + 13 c1 ρx = 0, σy  σ 2 i yy + 21 c1,y + 16 c21 − 12 c2 + 32 σy2 A ρy σy

ρxx − b2 ρx = 0, ρxy − ρyy − 3

h σyy ρyy − σy

3 2

+ c3 ρx + σy3 Bρ = 0.

(6.76)

A and B are determined through 2

σ   σ 2 yy yy +c1,y + 31 c21 − c2 + σy2 A = 0, − σy y σy

c1,yy + 2c1 c1,y − 3c2,y − 6b2 c3 + 49 c31 − 2c1 c2 + 6c4 + 3σy2 Ay − 6σy3 B = 0. (6.77) (2,2)

c) Janet basis of type J4,17 : σ = h(φ) where φ(x, y) = C is the solution of y 0 − 1 = 0, the system for ρ is b1 σ   σ  xx xx − b2 + 31 d1 ρy − b1 + 13 b1 d1 ρx = 0, σx σx   σ xx ρyy − b21 ρxx + 2b1 + 32 b1 d1 + c2 − 2b3 ρy σx   σ xx + 32 b21 d1 − b1 b3 ρx = 0, + 2b21 σx h  σ 2 σxx xx + 21 d1,x ρxxx − 3 ρxx + d2 ρy − 23 σx σx i + 16 d21 + 23 b1 d2 − 12 d3 + 32 σx2 A ρx + σx3 Bρ = 0. (6.78)

ρxy + b1 ρxx −

A and B are determined through     σxx 2 +d + 1 d2 − d + b d + σ 2 A = 0, 2 σσxx − 1,x 3 1 2 x σx 3 1 x x d1,xx + 2d1 d1,x − 3d3,x + 3b1 d2,x + 2d1,x d1 + 6d4 − 2d1 d3 +2b1 d1 d2 − 6b2 d2 − 3b1 b2 d2 + 49 d31 + 3σx2 Ax − 6σx3 B = 0. (6.79) Proof In case a) the given equations are obtained from a Janet basis (2,2) relating σ and ρ to the coefficients of the given type J4,2 for the symmetry generators. A(x) ≡ P (σ) and B(x) ≡ Q(σ) are the transformed coefficient functions P and Q. Their actual shape is determined by the choice of σ.

Transformation to Canonical Form

301

This freedom is allowed by the structure invariance group for the canonical form according to Theorem 6.30. The remaining four constants occur in the solution for ρ. The discussion for case b) is similar. Example 6.32 The equation considered in Example 5.43 generates the (2,2) type J4,2 Janet basis, therefore case a) of the above theorem applies. If σ = x is chosen, the system (6.74) yields the independent solution ρ = x y and the canonical form 8u 8 v 000 − v0 + v = 0. (6.80) 2u − 1 2u − 1 1 yields ρ = 1 , and the canonical form Choosing σ = x xy v 000 +

8 8 v0 − 5 v=0 u4 (u − 2) u (u − 2)

(6.81)

is obtained. Example 6.33 The equation considered in Example 5.44 generates the (2,2) type J4,19 Janet basis, therefore case b) of the above theorem applies. If σ = y is chosen, the system (6.74) yields the independent solution ρ = x y and 1 with the the canonical form (6.80). Choosing σ = y1 instead yields ρ = xy canonical form (6.81). Example 6.34 The equation considered in Example 5.45 generates the (2,2) type J4,17 Janet basis, therefore case c) of the above theorem applies. The first order equation for σ is y 0 +1 = 0; i.e., φ(x, y) = x+y. Choosing σ = x+y yields a system for ρ with the fundamental system {0, x, y}. Consequently, any linear combination ρ = c1 x+c2 y with c1 6= c2 yields a possible transformation to a canonical form which turns out to be (6.80). Five-Parameter Symmetries. The parameter a that defines the symmetry 3 class S5,2 (a) has been determined in Theorem 5.17. The expressions for P , Q and R occurring in the following theorem are also given there. The Janet bases of order five that are required are listed on page 382. 3 or Theorem 6.32 If a third order ode belongs to the symmetry class S5,1 3 S5,2 (a), three types of Janet bases may occur. The systems of pde’s for the transformation functions σ and ρ are as follows. (2,2)

a) Janet basis type J5,1 : σx =

P a2 − a + 1 , Q (a − 12 )(a + 1)(a − 2)

σy = 0,

(6.82)

302 1 3

  (a + 1)σx + 3a1 + e1 ρy = 0, ρyy − c1 ρy = 0,   ρxxx − (a + 1)σx + 3a1 ρxx + e2 ρy ρxy −

(6.83)

  + aσx2 + (a + 1)a1 σx + 2a21 − a1,x ρx = 0. (2,2)

b) Janet basis type J5,2 : a2 − a + 1 P , 1 Q (a − 2 )(a + 1)(a − 2)   ρxx − c1 ρx = 0, ρxy − 31 (a + 1)σy + 3b1 + e1 ρx = 0,   ρyyy − (a + 1)σy + 3b1 ρyy   + aσy2 + (a + 1)b1 σy + 2b21 − b1,y ρy + e3 ρx = 0. σx = 0,

σy =

(6.84)

(2,2)

c) Janet basis type J5,3 : P a2 − a + 1 , σy = −c1 σx , (6.85) Q (a − 21 )(a + 1)(a − 2)   ρxy + c1 ρxx − 31 (a + 1)σx + e1 + 3a2 ρy   − 31 c1 e1 − 3c21 a3 + c1 (a + 1)σx ρx = 0,   ρyy − c21 ρxx + 32 (a + 1)c1 σx + 23 e1 c1 − c3 − c21 a3 − b2 ρy   + 23 (a + 1)c1 σx + 32 e1 c1 − c3 − 2c21 a3 c1 ρx = 0,   ρxxx − (a + 1)σx − 3c1 a3 ρxx + e2 ρy   + aσx2 − (a + 1)c1 a3 σx + (a3,x + e2 + a3 a2 + 3c1 a23 )c1 ρx = 0. (6.86) σx =

Example 6.35 From the Janet basis coefficients in Example 5.46, the system for σ is σx = y, σy = x, i.e., σ = xy + C. For ρ the system 1 2 ρxy − x y ρxx + x ρy − y ρx = 0, 2 xy 2 + 3y xy + 3 ρyy − x2 ρxx + y1 ρy − x2 ρx = 0, ρxxx − yρxx + ρy − ρx = 0 y y x3 x2 y is obtained with the three rational solutions ρ = 1, ρ = xy and ρ = x . The last alternative yields a transformation withqnonvanishing functional determinant; √ i.e., the desired transformation is x = u v , y = uv, and the canonical form is v 000 − v 00 = 0.

Transformation to Canonical Form

303

Example 6.36 Choosing a = 3 from Example 5.47, case b) of the above theorem leads to σx = 0, σy = 1, i.e., σ = y + C. The system ρxx = 0, ρxy + y1 ρx = 0, 3xy 2 + 8xy + 6x ρyyy − 4ρyy + 3ρy + ρx = 0 y3 has the rational solution ρ = x y . Together with the special solution σ = y, it 000 yields the canonical form v − 4v 00 + 3v 0 = 0 for the ode of Example 5.47. Maximal Symmetry Group with Seven Parameters. The canonical form corresponding to this group does not contain any unspecified elements; its structure invariance group is therefore identical to its symmetry group. The proceeding is similar to the second order equations with projective symmetry. This case has been considered in detail by Zorawski [192]. Lemma 6.10 (Zorawski 1897)  The differential invariants up to order three for the group S73 generated by ∂σ , ∂ρ , σ∂σ , ρ∂ρ , σ∂ρ , σ 2 ∂ρ , σ 2 ∂σ + 2σρ∂ρ with σ ≡ σ(x, y) and ρ ≡ ρ(x, y) may be described as follows. σ a) For σx 6= 0 there is a single first order invariant I1 = σxy , there are four second order invariants I2 = I1,x , I3 = I1,y , I4 =

σx ρxy − σxy ρx + σxx ρy − σy ρxx σxx −2 , σx ρy − σy ρx σx

I5 =

σxy σx ρyy − σyy ρx + σxy ρy − σy ρxy −2 σx ρy − σy ρx σx

and there are eight third order invariants I6 =

 σ 2 σx ρxxx − σxxx ρx σxxx σx ρxx − σxx ρx σxx xx − 32 , I7 = −3 , σx σx σ x ρy − σ y ρx σx ρy − σy ρx σx

I8 = I4,x , I9 = I4,y , I10 = I5,y , I11 = I2,x , I12 = I2,y , I13 = I3,y . b) For σx = 0 there are no first order invariants, there are two second order invariants ρxx ρxy σyy J1 = , J2 = − ρx ρx σy and there are five third order invariants J3 =

σyyy 3  σyy 2 − , J5 = J1,x , J6 = J1,y , J7 = J2,y , σy 2 σy J4 =

ρyyy σy − σyyy ρy ρyy σy − σyy ρy . − 3σyy σ y ρx σy2 ρx

304 Proof The prolongations of the group generators up to order three are, after some reductions have been performed, ∂σ , ∂ρ , σx ∂σx + σy ∂σy + σxx ∂σxx + σxy ∂σxy + σyy ∂σyy +σxxx ∂σxxx + σxxy ∂σxxy + σxyy ∂σxyy + σyyy ∂σyyy , ρx ∂ρx + ρy ∂ρy + ρxx ∂ρxx + ρxy ∂ρxy + ρyy ∂ρyy +ρxxx ∂ρxxx + ρxxy ∂ρxxy + ρxyy ∂ρxyy + ρyyy ∂ρyyy , σx ∂ρx + σy ∂ρy + σxx ∂ρxx + σxy ∂ρxy + σyy ∂ρyy +σxxx ∂ρxxx + σxxy ∂ρxxy + σxyy ∂ρxyy + σyyy ∂ρyyy , σx2 ∂ρxx

+ σx σy ∂ρxy + σy2 ∂ρyy + 3σx σxx ρσxxx +(2σx σxy + σxx σy )∂ρxxy + (2σxy σy + σx σyy )∂ρxyy + 3σy σyy ∂ρyyy ,

ρx σ∂ρx + ρy σ∂ρy + σx2 ∂σxx + 2σx ρx ∂ρxx + σx σy ∂σxy + (ρx σy + ρy σx )∂ρxy +σy2 ∂σyy + 2ρy σy ∂ρyy + 3σxx σx ∂σxxx + 3(ρxx σx + ρx σxx )∂ρxxx +(2σxy σx + σxx σy )∂σxxy + (2σxy σy + σx σyy )∂σxyy +(2ρxy σx + ρxxy σ + ρxx σy + 2ρx σxy + ρy σxx )∂ρxxy +(2ρxy σy + σx σyy + ρyy σx + 2ρy σxy )∂ρxyy + 3σyy σy ∂σyyy +3(ρyy σy + ρy σyy )∂ρyyy . They generate a system of linear homogeneous pde’s in the derivatives of σ and ρ for the desired invariants. There are 4, 10 or 18 derivatives up to order 1, 2 or 3 respectively. If all derivatives are nonvanishing, the above system has rank three w.r.t. to derivatives of first order, and rank five w.r.t. derivatives of order higher than one. From these figures the number of invariants in case a) follows. Their explicit form is obtained by applying the methods described in Section 5 of Chapter 2. The occurrence of the derivatives of the various orders guarantees their functional independence. If some partial derivatives vanish as it occurs in case b), the corresponding terms in the system for the invariants disappear. Lemma 6.11 (Zorawski 1897) The coefficients of an equation with S73 symmetry may be expressed in terms of the invariants introduced in the above lemma. a) For equation (5.43) the expressions for the coefficients in terms of the I 0 s are: A0 = I1 , A1 = 3I1 I5 − 6I3 , A2 = 3I1 I4 + 3I5 − 6I2 , A3 = 3I4 , B1 = I15 I7 + I14 (2I6 + I11 + I42 ) + I13 (I12 + I8 + I4 I5 − I2 I4 ) + I12 (I13 + I32 − I3 I4 − I2 I5 ) − I1 (I10 + 3I3 I5 − I2 I3 ) + 3I32 ,

Transformation to Canonical Form

305

B2 = 5I14 I7 + I13 (8I6 + 5I11 + 5I42 ) + I12 (5I12 + 3I8 + 5I4 I5 − 5I2 I4 ) + I1 (2I13 − 2I9 + 2I52 − 5I3 I4 − 5I2 I5 + I22 ) − I10 − 3I3 I5 + 7I2 I3 , B3 = 10I13 I7 + 2I12 (6I6 + 5I11 + 5I42 ) + I1 (7I12 + 2I8 + 7I4 I5 − 10I2 I4 ) + I13 − 2I9 + I32 − 4I3 I4 − 4I2 I5 + 4I22 , B4 = 10I12 I7 + I1 (8I6 + 9I11 + 9I42 ) + 3I12 + 3I4 I5 − 6I2 I4 , B5 = 5I1 I7 + 2I6 + 3I11 + 3I42 , B6 = I7 . b) For equation (5.44) the expressions for the coefficients in terms of the J 0 s are: A1 = 3J2 , A2 = 3J1 , B1 = −J4 , B2 = −3(J22 + J7 + J3 ), B3 = −3(J1 J2 + J6 ), B4 = −J12 − J5 . Proof If the canonical form equation v 000 (u) = 0 is transformed by a general transformation u = σ(x, y) and v = ρ(x, y), the coefficients Aj and Bk of (5.43) or (5.44) are obtained in terms of σ, ρ and its derivatives depending on whether σx 6= 0 or σx = 0. On the other hand, the invariants have been expressed in terms of them in Lemma 6.10. If σ, ρ and its derivatives are eliminated between the two sets of equations, the above expressions for the coefficients are obtained. Theorem 6.33 The transformation of an equation with S73 symmetry to canonical form is determined by a system of partial differential equations the general solution of which contains seven constants. a) For equation (5.43), the transformation function σ is determined by the system Sx − 12 S 2 − I6 = 0,

Sy − 12 I1 S 2 − I1,x S − I1 I6 − I1,xx = 0,

σy − I1 σx = 0, σxx − Sσx = 0.

(6.87) (6.88)

The transformation function ρ is explicitly given by I 1  σ 2 ρ=− I7 ρ1 dx 2 σx 2   i h  σ (ρ1,xx − I1 I7 ρ1 ) − σσx 12 σσx σσxx + 1 ρ + ρ dy + 1 1,x 1 2 σx x I σ i σ σ h σx  xx +σ I7 2 dx + 2 ρ1,xx − + ρ1,x + I1 I7 ρ1 dy σx σ σx σx I h  i 1 σxx I7 ρ1 dx+ ρ1,xx − ρ1,x + I1 I7 ρ1 dy −σ 2 2 σx 2σx (6.89)

306 where ρ1 = exp

I 

I4 +

 σxx  σxx  dx+ I5 + I1,x + I1 dy. σx σx

The invariants occurring in these equations are expressed in terms of the coefficients of (5.43) by I1 = A0 , I4 = 13 A3 , I5 = 2A0,x + 13 (A2 − A0 A3 ), I6 = − 12 A3,x + 12 B5 − 16 A23 − 52 A0 B6 , I7 = B6 . b) For equation (5.44), the transformation function σ is determined by the system Sx = 0, Sy − 21 S 2 − J3 = 0,

σx = 0, σyy − Sσy = 0.

(6.90)

The transformation function ρ is explicitly given by I h  i σ  1 σ σyy 1  σ 2 ρ= − ρ1,yy + + 1 ρ1,y − ρ1 dx 2 σy σy 2 σy σy − 21 J4 I +σ −σ

2

I



σ σy

2

ρ1 dy

σ i σy  σ h σ yy ρ1,yy − + ρ1,y dx + J4 2 ρ1 dy 2 σy σ σy σy  i 1 h σyy ρ ρ dx + J ρ dy − 1,yy 1,y 4 1 σy 2σy2 (6.91)

where

I ρ1 = exp

J2 dx+



J2 +

σyy  dy. σy

The required invariants may be expressed in terms of the coefficients as J1 = 13 A2 , J2 = 13 A1 , J3 = − 12 B2 − 12 A1,y − 16 A21 , J3,x = 0, J4 = −B1 . is essentially the Proof Case a). The first equation (6.87) for S ≡ σσxx x definition of I6 in Lemma 6.10. In order to obtain the second equation, the σ relation σxy = I1 is differentiated twice w.r.t. x. In the resulting expression,
σxx is substituted by means of the first equation of (6.87), and the obvious σx x
σ σxy σ relation σσxx − σσxx is used to eliminate σxxy . = σxxy x y x x σx x Eliminating the ρ-derivatives from the expressions for I4 , I5 and I7 in Lemma 6.10 and using the equations (6.87) for eliminating derivatives of σσxx , x

Transformation to Canonical Form the linear homogeneous system   σxx  σxx  ρy + I1 I4 − I1,x + I1 ρx = 0, ρxy − I1 ρxx − I4 + σx σx  σxx  ρyy − I12 ρxx − I1 I4 + I5 + I1,x + 2I1 ρy σx  σxx  ρx = 0, + I12 I4 + I1 I5 − I1,y + 2I12 σx h σxx 3  σxx 2 i ρxxx − 3 ρxx − I7 ρy + I1 I7 − I6 + ρx = 0 σx 2 σx

307

(6.92)

for ρ is obtained. It has the solutions ρ = 1, ρ = σ and ρ = σ 2 as it may be verified by substitution. Consequently, the right factor    σ  σ Lclm ρx , ρy , ρx − σσx ρ, ρy − σy ρ , ρx − 2 σσx ρ, ρy − 2 σy ρ =  n o 2  3 σxx ρ ≡ ρ ρ − I − ρy − I1 ρx ≡ ρ1 , ρxxx − 3 σσxx 6 x 2 2 σx2 x xx (6.93) may be divided out yielding the exact quotient  o n  σxx )ρ ρ , ρ − (I + I + I ρ2 − I7 ρ1 , ρ1,x − I4 + σσxx 1 1,y 5 1,x 1 1 σx x from which ρ1 and ρ2 may be determined by integration. Substituting it into the right hand sides of the second line of (6.93) yields an inhomogeneous system for ρ with known solution of the homogeneous part. It has been shown in Exercise 6 of Chapter 2, see also Appendix A, how a special solution for it may be obtained in terms of integrals. Applying these results leads to the representation (6.89) for ρ. Case b). The function σ follows from the expression for the invariant J3 in σ Lemma 6.10, the substitution σyy = − 12 S and two subsequent integrations. y The expressions for J1 , J2 and J4 in the same lemma yield the system   σ ρxx − J1 ρx = 0, ρxy − J2 + σyy ρx = 0, y  2  σyy σyyy σ ρyyy − 3 σyy ρ − − 3 ρy − J 4 ρx = 0 yy σ y y σy2 for ρ with the three solutions ρ = 1, ρ = σ and ρ = σ 2 , see Lie [109], part III, page 409. Consequently, the right factor 2  n σ o σyy σyy yyy ρx ≡ ρ1 , ρyyy − 3 ρy ≡ ρ2 ρyy − −3 2 σy σy σy may be divided out yielding the exact quotient n  o σyy  ρ2 − J4 ρ1 , ρ1,x − J1 ρ1 , ρ1,y − J2 + ρ1 . σy

308 By similar methods as in case a) the representation (6.91) is obtained. The required invariants are obtained by elimination from the expressions for A0 , . . . , A3 , B6 and B7 in Lemma 6.11. In either case, determining σ requires solving a first order partial Riccatilike system and integrations. Although the expressions (6.89) and (6.91) for ρ appear somewhat awkward, they are in fact very simple. Containing only arithmetic and integrations, they are guaranteed to be Liouvillian over the field determined by σ and ρ1 . Example 6.37 Equation 7.8 of Kamke’s collection in symmetry class S73 has been considered in Example 5.48 of the preceding chapter. From its 3 are obtained. coefficients the invariants I1 = I4 = I6 = I7 = 0 and I5 = − 2y They yield Sx − 21 S 2 = 0, Sy = 0 for S. The special solution S = 0 leads to σy = σxx = 0 with the non-constant solution σ = x. Substituting it together 1 into (6.89) yields ρ = √2 − 2. with ρ1 = √ y y y Example 6.38 An equation in symmetry class S73 , case a), has been considered in Example 5.49 of the preceding chapter. From the coefficients given there the invariants I1 =

1 1 3y x , I4 = − , I5 = − , I6 = 0, I7 = − 3 y x y x

are obtained. Substituting I1 and I6 into (6.87) yields Sx − 21 S 2 = 0, x S 2 − 1 S = 0 with the only rational solution S = 0. It leads to Sy + 2y y σy − x σ = 0, σxx = 0 a special solution of which is σ = xy. Substituting y x 1 into (6.89) finally yields this choice of σ and the corresponding value ρ1 = x y ρ = x. Example 6.39 An equation in symmetry class S73 , case b), has been considered in Example 5.50 of the preceding chapter. From the coefficients given there the invariants J1 =

x2 − 2x + 2 3 6x , J2 = , J3 = 0, J4 = − 3 x(x − 1) y y (x − 1)

are obtained. Substituting J3 into (6.90) yields Sy − 12 S 2 = 0 with the solution 2 . Choosing C = 0 leads to σ + 2 σ = 0 a special solution of S = −y + yy y y C which is σ = y1 . Substituting this choice of σ and the corresponding value (x − 1)y x y ρ1 = e into (6.91) finally yields ρ = x ex . x2

Transformation to Canonical Form

309

Exercises Exercise 6.1 Determine the constraints for the parameters and undetermined functions occurring in the canonical forms of Theorem 6.4 guaranteeing the correct symmetry type. Show that these constraints are related to the structure of the symmetry algebras. Exercise 6.2 Determine the transformation between the two canonical forms given in Examples 6.7 and 6.8. Exercise 6.3 Express the Janet basis coefficients of the two parameter symmetry groups in terms of the invariants determined in Lemma 6.1 and 6.2. Show that this is an application of Theorem 6.6. Exercise 6.4 Apply Theorem 6.7 to equation y 00 + r(y, y 0 ) = 0 in order to generate the most general canonical form as given in Theorem 6.4. Exercise 6.5 Consider again the equation discussed in Examples 5.15 and 6.12. Assume now that only a two-parameter group of type g26 is known. Determine the canonical form under this assumption and discuss the result. Exercise 6.6 Consider again equation 6.133 from Kamke’s collection discussed in Example 6.11. Assume now that only a two-parameter group of type g25 is known. Determine the canonical form under this assumption and discuss the result. Exercise 6.7 The number of subgroups that may be applied for generating a canonical form of a second order equation is maximal for symmetry class S82 . Reconsider equation 6.180 from Kamke’s collection that has been discussed in Example 6.14. Choose the two-parameter subgroups of type g25 and g26 and generate the respective canonical form. What about subgroups comprising three or four parameters?

Chapter 7 Solution Algorithms

It is assumed now that for a given ode of order not higher than three with nontrivial Lie symmetries the transformation functions from actual variables x and y(x) to canonical variables u and v(u) are known. This implies that its infinitesimal symmetry generators are explicitly known in canonical variables from the tabulation in Section 5.1 without further calculations. What remains to be done is to solve the canonical equation, and to generate the solution in actual variables from it. The term solution always means general solution; i.e., a n-parameter family of solutions if n is the order of the equation. This excludes the so-called singular solutions which may or may not be specializations of the general solution. The relation between these two concepts is discussed in detail in the article by Buium and Cassidy [23], Section 1.8, and articles by Ritt quoted there. Before this proceeding is described in detail for the individual symmetry classes, the underlying general principles will be outlined. The following discussion is based on Engel [44], vol. V, Anmerkungen by Engel, pages 643-669 and 682-687 where many more details may be found. Let the canonical ode of order n have an r-parameter symmetry group. The further proceeding depends crucially on the relative values of r and n. At first it is assumed that r < n and ∆ 6= 0; the Lie determinant ∆ is defined by (5.15). The fundamental invariants Φ1 and Φ2 defined on page 201 are of order r − 1 and r respectively. In terms of these invariants the canonical equation has the form  dΦr dn−r Φr  Ω Φr−1 , Φr , ,..., =0 (7.1) dΦr−1 dΦn−r r−1 where Ω is an undetermined function. This is a general ode of order n − r. It has to be solved first. If this is not possible the main achievement of the symmetry analysis consists in lowering the order of the originally given ode of order n to an equation of order n − r in the independent variable Φr−1 and dependent variable Φr . If the solution of (7.1) may be found, however, an equation of order r allowing an r-parameter symmetry group is obtained. Consequently, the case n = r has to be considered next. It is discussed in full detail for arbitrary values of n in the above mentioned Anmerkungen by Engel. The complete answer for n = 2 and n = 3 that is relevant here is given in the book by Lie [113] on differential equations, Theorem 45 on page 464 and Kapitel 25, in particular Theorem 49 on page 555. It may be formulated as follows.

311

312 Theorem 7.1 (Lie 1891) If a second order ode has a two-parameter symmetry group and its symmetry generators are explicitly known, its solution involves two integrations. If a third order ode has a three-parameter solvable symmetry group with known generators, its solution involves three integrations. If the group is not solvable, the solution of a first order Riccati equation is required. The proof is based on the representation of the given ode as an equivalent first order pde, and Theorem 2.23 dealing with its solution. The details may be found in Lie’s book. If n < r, i.e., if the number of symmetries is higher than the order of the differential equation, the solution process may be further simplified. In many cases the solution may be given explicitly without any integrations involved. The complete answer for arbitrary values of n and r including all exceptions is due to Engel [44]; see in particular pages 651-665. Engel’s comprehensive proof is based to a large extent on a case-by-case discussion. Therefore it is not given here. Rather for order n = 2 and n = 3 all possible symmetry types with r ≥ 3 or r ≥ 4 respectively are treated in detail below. For first order equations the situation is different because point symmetries cannot be determined algorithmically. Therefore, at least a single nontrivial symmetry generator must be known a priori in order to apply the algorithms described in this chapter. Alternatively, the symmetry problem may be specialized to certain classes of equations for which algorithmic methods are available, e. g. for Abel’s equation. This is discussed in Section 7.1. Subsequently, equations of order two and nonlinear equations of order three are treated in Sections 7.2 and 7.3 respectively. The subject of the final Section 7.4 are equations of order three that are equivalent to linear ones.

7.1

First Order Equations

In this section it is assumed that in addition to the first order equation to be solved at least a single symmetry generator is explicitly known; it is considered as part of the input to the solution algorithms. One-Parameter Symmetry. If for a quasilinear first order equation ω ≡ y 0 + r(x, y) = 0

(7.2)

a single nontrivial symmetry generator is known, it may be possible to introduce new variables u and v(u) according to Theorem 6.1 such that the symmetry transformation becomes the translation in v. In these variables equation (7.2) takes the form v 0 + s(u) = 0.

(7.3)

Solution Algorithms Integration yields the representation Z v + s(u)du = C

313

(7.4)

of its general solution. If the actual variables are resubstituted into (7.4), the general solution of (7.2) is obtained. The function field for its representation is determined by the integral in (7.4) and the transformation functions between the actual variables x, y and the variables u, v. According to Lemma 3.4 it is not guaranteed that the canonical form (7.3) from the knowledge of a single symmetry generator can be obtained. If this is not possible, the following lemma which is also due to Lie [113], Kapitel 6, §1, may be applied. Lemma 7.1 If a first order ode y 0 + r(x, y) = 0 allows the nontrivial 1 symmetry generator U = ξ(x, y)∂x + η(x, y)∂y , the expression is an η + rξ integrating factor. Consequently, there holds I rdx + dy . φ= (7.5) rξ + η

Proof Let φ(x, y) = C be the general solution of y 0 + r(x, y) = 0. By definition of a symmetry generator, it obeys ξφx + ηφy = 1. Moreover, as a solution, φ has to satisfy φx + y 0 φy = φx − rφy = 0. Eliminating the first r ,φ = 1 . order derivatives from these two expressions yields φx = η + rξ y η + rξ Substituting them into the right hand side of dφ = φx dx + φy dy yields dφ = dy + rdx 1 which is a total differential by construction. Consequently, η + rξ η + rξ is an integrating factor and the representation is obtained. The various steps described above are applied in the subsequent algorithm for solving equation (7.2) with a known symmetry generator. Algorithm 7.1 LieSolve1.1(ω, U ). Given a quasilinear first order ode ω in the form (7.2) with the known symmetry generator U , its general solution is returned. S1 : Determine transformation functions. Applying Lemma 3.4 determine the transformation of ω to canonical form (7.3). If it cannot be determined return goto S4. S2 : Canonical form. Applying the result of step S1, generate the canonical form (7.3). S3 : Return solution. Integrate the canonical form. Resubstitute the actual variables and return the result. S4 : Use integrating factor. Evaluate the integral in (7.5) and return the result.

314 Example 7.1 For equation 1.15 from the collection by Kamke the canonical form v 0 + 2 1 = 0 has been obtained in Example 6.1 by the transformau −1 −1 tion x = v and y = u + v 2 . It may be integrated to v + 12 log u u + 1 = C. 2 Resubstituting the actual variables by u = y − x and v = x finally yields y=

C(x2 − 1)e−2x − x2 − 1 . Ce−2x − 1

Two-Parameter Symmetries. If for a first order equation (7.2) the generators of a two-parameter symmetry group are explicitly known, its general solution may often be obtained by algebraic operations as the next result shows. Let an equation (7.2) have two symmetries with generators η + rξ2 Ui = ξi (x, y)∂x + ηi (x, y)∂y for i = 1, 2, and define F ≡ 2 . If ∂F 6= 0 η1 + rξ1 ∂x and ∂F 6= 0, its general solution is F (x, y) = C. ∂y Theorem 7.2

Proof The symmetry type g26 is considered first, i.e., [U1 , U2 ] = 0. Let the canonical generators in variables u and v ≡ v(u) be V1 = ∂u and V2 = ∂v . By Lemma 3.3 the transformation functions x = φ(u, v) and y = ψ(u, v) obey the relations V1 φ = φu = ξ1 , V2 φ = φv = ξ2 , V1 ψ = ψu = η1 , V2 ψ = ψv = η2 η1 + η2 v 0 is obtained. Substitution into ξ1 + ξ2 v 0 η + rξ2 (7.2) and some simplifications yields v 0 + 2 = 0. According to Theη1 + rξ1 orem 6.2 the canoncial form for this symmetry class is v 0 + a = 0. This is assured if the relation given above holds. The symmetry type g25 is considered next, i.e., now [U1 , U2 ] = U1 . From the canoncial generators V1 = ∂v and V2 = u∂u + v∂v , the relations from which the first derivative y 0 =

V1 φ = φv = ξ1 , V1 ψ = ψv = η1 , V2 φ = uφu + vφv = ξ2 , V2 ψ = uψu + vψv = η2 follow from which the first order derivatives η2 − vη1 ξ2 − vξ1 , φv = ξ 1 , ψ u = , ψv = η1 φu = u u η2 − η1 v + η1 uv 0 for ξ2 − ξ1 v + ξ1 uv 0 the first derivative. Substitution into (7.2) and rearranging terms yields now η + rξ2 v0 u − v + 2 = 0. This represents the correct canonical form v 0 + a = 0 η1 + rξ1 η + rξ2 η + rξ2 if 2 − v is proportional to u; i.e., if there holds 2 = v + au = C. η1 + rξ1 η1 + rξ1 of φ and ψ are obtained. They lead to the expression y 0 =

Solution Algorithms

315

If F defined in Theorem 7.2 is identically a constant, the above theorem cannot be applied for the solution procedure. In this case the canonical form v0 + a = 0

(7.6)

has to be determined by Theorem 6.2 from which the solution may always be obtained. Lie [113] discusses the two alternatives in detail on page 124, Satz 1. On pages 126 and 127 he explains the different behavior by geometrical considerations. A solution scheme based on the preceding discussion may be designed as follows. Algorithm 7.2 LieSolve1.2(ω, U1 , U2 ). Given a quasilinear first order ode ω in the form (7.2) with two known symmetry generators U1 and U2 , its general solution is returned. S1 : Generate F. From the coefficients of U1 and U2 generate F defined in Theorem 7.2. If F depends explicitly on x and y, return it. S2 : Return result. Generate the canonical form (7.6). Determine the solution in actual variables from it and return the result. It should be noted that due to the explicit knowledge of two symmetry generators this algorithm never fails. (x − y)2 + 1 = 0 with the two (x − y)2 − 1 symmetry generators U1 = ∂x + ∂y and U2 = y∂x + x∂y of a type g25 group. Substituting the coefficients of these generators into the definition of F in Theorem 7.2 yields after some simplifications the general solution in the form of (x + y + C)(x − y) = 1. Example 7.2

Consider the equation y 0 +

Example 7.3 The same equation as in the preceding example is considered. If only the generator U = ∂x + ∂y is known, the solution scheme of algorithm LieSolve1 has to be applied. By Lemma 3.4 the canonical variables 2 are obtained as x = v − u, y = v with canonical equation v 0 − u +2 1 = 0. 2u Integration and resubstitution of the original variables yields the same solution as above. However, knowing only a single symmetry generator requires an additional integration. Abel’s Equation. If one or two symmetry generators of an Abel equation may be found, its solution may be obtained by applying the above results for general first order equations. However, due to the special structure, the more explicit answer given in the subsequent theorem is usually more convenient. Theorem 7.3 The two rational normal forms of Theorem 4.14 are considered separately. i) If the rational normal form is y 0 + Ay 3 + By = 0, the general solution is y=q R ¯ 2B(

1 A ¯ dx B

(7.7) + C)

316 R ¯ = exp (2 Bdx) and C is the integration constant. where B ii) If the rational normal form is y 0 + Ay 3 + By + 1 = 0 and allows a one-parameter symmetry with generator  1 A0 y∂ y , 3A A1/3 R its canonical variables are u = A1/3 y and v = A1/3 dx. They lead to the canonical form 1 v0 + 3 =0 (7.8) u + ku + 1 √ where k = 3 K, and K is the absolute invariant (4.51). From (7.8) its general solution is obtained by integration as Z du v+ = C. (7.9) u3 + ku + 1 U=

1



∂x −

If k = 0 the integral may be performed with the result Z du 1 u+1 1 2u − 1 v+ = v + log p + √ arctan √ = C. (7.10) 2 3 u3 + 1 3 3 u −u+1 Proof The transformation functions u ≡ σ(x, y) and v ≡ ρ(x, y) obey the system σx ξ + σy η = 0 and ρx ξ + ρRy η = 1. With the above values for ξ and η the solution σ = A1/3 y and ρ = A1/3 dx follows. The first derivative ρ − σx v 0 y 0 is transformed according to y 0 = − x (compare (B.6) and (B.3)) ρy − σ y v 0 0 with the result y 0 = 10 − 31 A A y. Substituting this expression into the rational v normal form yields after some simplifications (7.8). The various alternatives that may occur for solving an Abel equation according to this theorem are combined in the following algorithm. The fundamental difference compared to LieSolve1.1 and LieSolve1.2 is the fact that it is not required to know any symmetry generator in advance. Rather it is automatically determined whenever this is possible. Algorithm 7.3 SolveAbel(ω). Given an Abel equation ω, its general solution is returned if a symmetry may be found, or f ailed otherwise. S1 : Second kind equation? If ω is an Abel equation of second kind, apply the transformation (4.40). S2 : Rational normal form. Transform ω to rational normal form. S3 : Bernoulli case? If the result of S2 is a Bernoulli equation, determine its solution from (7.7) and goto S5. S4 : Symmetry? If K(A, B) = constant, determine its solution from (7.9) or (7.10). Otherwise return f ailed.

Solution Algorithms

317

S5 : Backtransformation. Apply the reverse transformations performed in S1 and S2 and return the result. Next to Riccati’s equation, Abel’s equations are the most thoroughly studied differential equations in the history of mathematics. In the course of time several integrable cases have been found by some kind of heuristics, a short discussion of them may be found in Kamke [85], pages 24-28. It turns out that most of them may be subsumed under the symmetry analysis described here. Scalizzi (1917) describes the integrable case of (4.38) for the constraint a0 =

2 a32 1  a2 0 1 a1 a2 + . − 3 a3 3 a3 27 a23

From the last relation in (4.43) it is obvious that this condition identifies the rational normal form allowing a two-parameter symmetry with the solution (7.7). a3 )0 = Chiellini (1931) found an integrable case of (4.38) for a0 = a1 = 0, ( a 2 const. · a2 and a2 6= 0, a3 6= 0. Under these constraints the coefficients a2 a3 (4.43) are b3 = a3 , b1 = − 3a2 and b0 = const. · 22 ; i.e., due to b0 6= 0 a 3 a3 rational normal form corresponding to case ii) of Theorem 4.14 is obtained. a0 a2 a0 Applying the relation a33 − a22 = const. · a23 which is a consequence of the above constraints it follows that the absolute invariant K(A, B) is constant. Consequently, there is a one-parameter symmetry group with generator U=

 a0 i a3 h a03  2 ∂ − 2 − y∂ . y x a22 a2 a3

Chini (1924) observed that the condition Φ3 = 0 for the invariant defined in Theorem 4.15 leads to a Bernoulli equation. Evaluating it for the equation with coefficients bk in the proof of Theorem 4.14 yields Φ3 = 3b0 b23 . Because b3 6= 0 it follows b0 = 0, i.e., the Bernoulli rational normal form. For Φ3 6= 0 Chini showed that under the condition 5/3

a3 Φ03 + (3a1 a3 − a22 − 3a03 )Φ3 = 3αΦ3

the integral representations described in case ii) of Theorem 7.3 exist. Evaluating the above constraint for the rational normal form of this case shows that it is equivalent to the condition K(A, B) = const.; i.e., the corresponding equation allows a symmetry generator. For the equation of the second kind (4.39) Abel himself described several integrable cases if certain relations between the coefficients are satisfied; e.g., if f1 = 2f2 − g 0 the transformed equation becomes y 0 + (f0 − f2 g 2 + gg 0 )y 3 + f2 y = 0

318 which is a Bernoulli equation. Finally Todorov and Kristev [177] consider an Abel equation of the second kind (4.39) and impose the constraints f0 = (g 3 − 1)f3 + gg 0 ,

f1 = 3f3 g 2 − g 0 ,

f2 = 3f3 g.

The signs of the fk ’s have been reversed in order to agree with the notation f0 in (4.39). They imply the rational normal form y 0 + f32 y 3 + 3 y + 1 = 0 with a f3 vanishing absolute invariant and the integral representation (7.10) (equation (14) in Todorov and Kristev [177]). Example 7.4 For equation 1.46 of Kamke’s collection the rational normal form z 0 − x2 z 3 + 22 z = 0 has been obtained in Example 5.7. The solution of x this Bernoulli equation is obtained from (7.7) in step S3. Resubstituting the original variable z = y − 12 yields the final answer x 1 e2/x + 2. y=q R x −2 x2 e4/x dx + C It is not always advisable to generate the canonical form (7.8) because √ it may lead to unnecessary algebraic numbers due to the cubic root in k = 3 K. The next example shows how they may be avoided. 3
y 0 + 1 y 2 − 1 y = 0 from Kamke’s Example 7.5 Equation 1.255 y − x x x2 collection is an Abel equation of the second kind. Introducing the new depen3 (compare equation (4.39) and the transformation dent variable z by y = z1 + x 1 z = 0. It is transformed into rational 6 0 given there) leads to z − 3 z 3 − 22 z 2 − x x x 3 32 w 7w 1 normal form w0 − 19683 3 − 9 x + 1 = 0 by z = − 243 (4w + 27x). Its absolute x √ 9 3 invariant is K = − 729 4 ; consequently, k = − 2 2. In order to avoid the cubic 2y root, it is better to introduce canonical variables by u = 9x and v = − 92 log x with inverse x = exp (− 29 v) and y = 29 u exp (− 92 v). It yields v0 +

4 3 27 u

27 1 4 = v0 + =0 3 2 (u + 2 ) (u − 3) −u−1

3 +3 with the solution v = 13 log 2u u − 3 − 2u + 3 + C. Resubstituting the original variables w, z and finally y yields the solution of the originally given equation in the form xy 3 = C exp (xy).

Solution Algorithms

7.2

319

Second Order Equations

The subjects of this section are second order equations that are linear in the highest derivative. A cursory glance at the various collections of solved examples shows that they have been investigated more thoroughly than any other family of equations. Lie [113] himself devoted to them almost completely his book on symmetries of differential equations. These results are discussed in detail in the current section. One-Parameter Symmetry. If a second order quasilinear ode belongs to symmetry class S12 it is equivalent to an equation of the form v 00 + r(u, v 0 ) = 0

(7.11)

with r(u, v 0 ) an undetermined function of its arguments. The substitution w = v 0 in (7.11) yields w0 + r(u, w) = 0. (7.12) Because there are no constraints on r(u, w), this is a general first order equation. It is the first example of an equation of the form (7.1) because the number of symmetries is lower than the order of the equation. As explained in the introduction to this chapter, solving it is a new problem that has to be handled as described in the preceding Section 7.1. The further proceeding depends on whether or not this first order equation can be solved. The algorithm below applies these steps. The solution procedure may terminate in step S2 if σ cannot be obtained, or in step S3 if equation (7.12) cannot be solved. Algorithm 7.4 LieSolve2.1(ω). Given a quasilinear second order ode in symmetry class S12 , its general solution or an equivalent first order equation is returned; or failed if neither may be found. S1 : Janet basis for symmetries. Generate a Janet basis for the determining system of its Lie symmetries. S2 : Determine transformation functions. If case a) of Theorem 6.7 applies choose σ. If case b) applies solve the first order equation for σ. If it cannot be obtained return failed. Determine ρ by integration. S3 : Canonical form. Applying σ and ρ obtained in S2 generate the canonical form (7.12) and apply the solution algorithms of Section 7.1 to it. If a solution has not been found, return the canonical form equation together with the transformation functions σ and ρ. S4 : Return solution. From the solution obtained in step S3, construct the solution of ω and return the result. Example 7.6 For equation 6.90 of Kamke’s collection the canonical form has been obtained in Example 6.5. Substituting v 0 = w yields w0 + u(u + 5)w3 − (u + 3)w2 + 14 w = 0.

320 This is an Abel equation the solution of which cannot be obtained. Example 7.7 The canonical form (6.2) for equation 6.98 of Kamke’s collection leads to w0 + 2(2u − 1)w2 + w = 0. This Bernoulli equation has the Z du 1 general solution w = and finally v = + C2 . C1 ex − 4x − 2 C1 eu − 4u − 2 y If the actual variables u = 2 and v = log x are resubstituted, an integral x representation of the general solution of equation 6.98 is obtained. Two-Parameter Symmetries. The canonical form of a second order quasi2 linear ode in the symmetry class S2,1 has been shown to be v 00 + r(v 0 ) = 0

(7.13)

where r(v 0 ) is an undetermined function of v 0 . In order to solve this equation it is rewritten as v 00 + r(v 0 ) =

dv 0 dv 0 + r(v 0 ) = 0 or + du = 0. du r(v 0 )

(7.14)

Integration yields R(v 0 ) + u = C1

with

R(v 0 ) =

Z

dv 0 . r(v 0 )

(7.15)

If r(v 0 ) is such that the latter integration may be performed, the original equation is reduced to a first order equation. Depending on the form of R(v 0 ) it may be possible to continue the solution procedure. In any case the general solution may always be represented in parameter form as Z u + R(p) = C1 , v + pR(p) = R(p)dp + C2 . (7.16) This is a consequence of the fact that (7.15) is a first order differential equation that does not contain the dependent variable explicitly; i.e., it allows the symmetry ∂v . Details may be found in Kamke [86], vol. I, page 49; see also Exercise 7.4. If u ≡ σ(x, y) and v ≡ ρ(x, y) are resubstituted, the solution in actual variables is obtained. In the subsequent algorithm the various steps for solving an equation in 2 symmetry class S2,1 are put together. Only step S2 may fail if the transformation to canonical form is not Liouvillian. Algorithm 7.5 LieSolve2.2.1(ω). Given a quasilinear second order ode 2 ω in symmetry class S2,1 , its general solution is returned explicitly or in parameter form. Or failed if it cannot be determined. S1 : Equations for σ and ρ. Generate a Janet basis for the symmetry generators of ω and set up system (6.3).

Solution Algorithms

321

S2 : Determine transformation functions. Determine an independent pair of Liouvillian solutions of the system obtained in step S1. If it cannot be found return failed. S3 : Generate canonical form. By means of the transformation functions obtained in S2 generate the canonical form (7.13). S4 : Return solution. Using r(v 0 ) from step S3 construct the solution (7.16), resubstitute actual variables and return the result. Example 7.8 In the preceding chapter, a canonical form of Kamke’s equation 6.227 has been determined in Example 6.7. Substituting the respective value for r(v 0 ) into (7.15) and (7.16) and performing the integrations yields u + log

p p+2 = C¯1 , + p+1 p+2

v + log (p + 1) −

2p = C¯2 . p+2

If the canonical variables are resubstituted, the parameter representation  p p  p+1 1 x = C1 exp , y = C2 exp − p+2 p+2 p+2 p+2 for x and y is obtained. In Exercise 7.6 the same answer will be obtained from the canonical form of Example 6.8. 2 The other two-parameter symmetry class is S2,2 . A quasilinear second order ode in this symmetry class has the canonical form uv 00 + r(v 0 ) = 0

(7.17)

where again r(v 0 ) is an undetermined function of v 0 . Rewriting it as v 00 +

dv 0 r(v 0 ) dv 0 du r(v 0 ) = + = 0 or + =0 u du u u r(v 0 )

(7.18)

it may be integrated once with the result R(v 0 ) + log u = C1

with R(v 0 ) =

Z

dv 0 . r(v 0 )

(7.19)

The same general remarks apply to this equation as for (7.15). If a further integration is possible, the general solution of the canonical equation follows explicitly. Otherwise the parameter representation Z u = C1 e−R(p) , v = C1 (pe−R(p) − e−R(p) dp) + C2 (7.20) in canonical variables or, after substitution of σ(x, y) and ρ(x, y) in actual variables is obtained. As in the previous case, details may be found in Kamke [86], vol. I, page 49.

322 2 A solution scheme for equations in symmetry class S2,2 is realized in terms of the following algorithm.

Algorithm 7.6 LieSolve2.2.2(ω). Given a quasilinear second order ode 2 ω in symmetry class S2,2 , its general solution is returned explicitly or in parameter form. S1 : Equations for σ and ρ. Generate a Janet basis for the symmetry generators of ω. S2 : Determine transformation functions. Determine the transformation functions σ from (6.6), and ρ from (6.7) or (6.8). S3 Generate canonical form. By means of the transformtion functions obtained in S2 generate the canonical form (7.17). S4 : Return solution. Construct the solution (7.20), resubstitute the actual variables and return the result. According to Theorem 6.3, for any equation in this symmetry class a canonical form may be obtained. Consequently, a solution in parameter form is guaranteed and the above algorithm LieSolve2.2.2(ω) does not have a failed exit. In those cases where (7.19) may be solved for v 0 , the solution may be expressed as an integral as it is the case in the following example. Example 7.9 The canonical form for Kamke’s equation 6.159 has been determined in Example 6.9. With the above notation there follows r(v 0 ) = − 23 v 0 (v 02 − 1) and R(v 0 ) =

2 3

log v 0 −

1 3

log (v 02 − 1).

Substituting it into (7.20) leads to u3 v 02 = C1 (v 02 − 1). Solving for v 0 and integrating leads to Z du p v = C1 + C2 . 3 u + C12 If the actual variables u = √1 and v = x are resubstituted, and the integray tion constant C1 is replaced by its inverse, the solution of the original equation is obtained in the form Z dy 1 q q x+ 2 = C2 . √ √ y y y y + C1 Three-Parameter Symmetries. In this subsection the number of symmetries exceeds the order of the respective ode by one and the remarks of the 2 second paragraph on page 312 apply. In the first symmetry class S3,1 to be discussed the canonical form is √
v 00 (u − v) + 2v 0 v 0 + a v 0 + 1 = 0 (7.21) where a is a constant, a 6= 0, or in rational form v 002 (u − v)2 + 4v 00 v 0 (v 0 + 1)(u − v) + 4v 04 + 4(2 − a3 )v 03 + 4v 02 = 0. (7.22)

Solution Algorithms

323

Two subsequent integrations of the canonical form (7.21) yield √ 1 (u − v) v 0 √ = 0 C1 1 + a v0 + v and finally a third integration leads to the general solution (C1 v + C2 )(C1 u + C2 + a) + 1 = 0.

(7.23)

The various steps for solving an equation in this symmetry class are put together in the following algorithm. Algorithm 7.7 LieSolve2.3.1(ω). Given a quasilinear second order ode 2 ω in symmetry class S3,1 , its general solution is returned; or failed if it cannot be determined. S1 : Equations for R and S. Generate Janet basis for the symmetry generators of u. Set up system (6.13), (6.16) or (6.18) respectively for R, or system (6.20) for R and S. S2 : Determine R and S. From the equations obtained in S1 determine rational solutions for R and S. If none can be found return failed. S3 : System for transformation functions. Applying the result of S2 set up the system (6.14), (6.17), (6.19) or (6.21) for σ and ρ. S4 : Determine transformation functions. Determine two independent Liouvillian solutions for the system obtained in S2. S5 : Return solution. Generate the canonical form and determine the value for a from it. Substitute a and the canonical variables into (7.23) and return the result. Example 7.10 An equation in this symmetry class has been considered before in Examples 5.13 and 6.10. Comparing the coefficients of the cubic power of v 0 in the canonical form of latter example and of (7.22) yields a = 1. Substituting this value and the expressions for the canonical variables u = xy 2 into (7.23) yields the solution, after some simplifications, in the and v = x form y=

(C22 + C2 + 1)x + 2C1 (C2 + 1) . (2C1 + C2 x)C1 x

2 A quasilinear second order ode in symmetry class S3,2 is equivalent to an equation of the form v 00 v 3 + a = 0 (7.24)

where a is constant, a 6= 0. In order to
obtain the solution, the canonical 0 equation (7.24) is written as (v 02 )0 − a2 = 0. In this form the integration v p C1 v 2 + a . is obvious and leads to v 02 − a2 = C1 . Solving for v 0 yields v 0 = v v Separation of variables allows it to perform the second integration with the result vdv 1 p p = du or C1 v 2 + a − u = C2 . C1 C1 v 2 + a

324 The latter relation may be rewritten as C1 v 2 − C12 (u + C2 )2 + a = 0.

(7.25)

Similar to the preceding case the solution procedure comes down to determining the transformation functions of the actual to canonical variables, determining the constant a from the canonical form and substituting it together with u ≡ σ and v ≡ ρ into the solution of the latter. Theorem 6.11 explains how the transformation to canonical variables is achieved. These steps are put together in the subsequent algorithm. Algorithm 7.8 LieSolve2.3.2(ω). Given a quasilinear second order ode 2 ω in symmetry class S3,2 , its general solution is returned; or failed if it cannot be determined. S1 : Equations for R. Generate Janet basis for the symmetry generators (2,2) (2,2) of ω. If it is of type J3,4 set up system (6.22), of type J3,7 system (2,2)

(6.23) and of type J3,6

system (6.24) for R.

S2 : Determine R. From the equations obtained in S1 determine a rational solution for R. If it cannot be found return failed. S3 : System for transformation functions. Applying the result of S2 set up the systems (6.22), (6.23) or (6.24) respectively for σ and ρ. S4 : Determine transformation functions. Determine two independent Liouvillian solutions for the system obtained in S3. S5 : Return solution. Generate the canonical form and determine the value for a from it. Substitute a and the canonical variables into (7.25) and return the result. Example 7.11 The solution procedure for equation 6.133 of Kamke’s collection may now be completed. Substituting u = x − y, v 2 = x + y and a = 41 as obtained in Example 6.11 into (7.25) yields the general solution in actual variables C1 (x + y) + C12 (x − y + C2 )2 +

1 4

= 0.

2 S3,3 (c)

For an equation in symmetry class it is assumed that the parameter c has already be determined. Its canonical form is v 00 + av 0(c−2)/(c−1) = 0

(7.26)

with a constant a 6= 0. In order to yield an integer power v 0m , the parameter c must be determined by c = 1 + 1 −1 m . For |m| ≤ 5 the resulting values of c are given in the following table. m -5 -4 -3 -2 -1 0 2 3 4 5 c 7/6 6/5 5/4 4/3 3/2 2 0 1/2 2/3 3/4

Solution Algorithms

325

Integrating the canonical form (7.26) once yields (c − 1)v 01/(c−1) + au = C1 . The second integration gives the final result v=

(1 − c)1−c (au − C1 )c + C2 . ac

(7.27)

This form is most suitable for applications. The value of the constant c determines the symmetry type completely. In Theorem 6.12 it is expressed in terms of the coefficients of the Janet basis for the symmetries. Combining these results the following algorithm is obtained. Algorithm 7.9 LieSolve2.3.3(ω). Given a second order quasilinear or2 dinary differential equation in symmetry class S3,3 (c), the general solution is returned. S1 : Janet basis for symmetries. Generate a Janet basis for the symmetry generators of ω. S2 : System for transformation functions. If the Janet basis obtained in (2,2) step S1 has type J3,6 , set up system (6.25), (6.26) or (6.27) respectively. (2,2)

If it has type J3,7 set up system (6.28). S3 : Determine transformation functions. Determine two independent Liouvillian solutions for the system obtained in S2. S4 : Return solution. Generate the canonical form (7.26), substitute a and the canonical variables into (7.27), and return the result. Example 7.12 Two equations with this symmetry group have been considered in Example 6.12. In both cases the canonical form is v 00 + v 04 = 0 corresponding to γ = 32 and a = 1. Substituting these values into (7.27) yields 2 3 v= √ (u − C1 ) 3 + C2 3 2 3 for the solution in canonical variables. Inverting the substitution functions 1 for the first alternative leads to the solution x = v1 and y = u   (8C23 − 9C1 )x3 + 24C22 x2 + 24C2 x + 8 y 2 − 18C1 x3 y − 9x3 = 0. In the other case, inverting x = v − u and y = 3u − 2v yields the solution
  1 y 3 − 9 x + 13 C1 − 18 y 2 + 27 x2 + ( 23 C1 − 16 )x + 19 C12 − 12 C2 y   1 +27 x3 + (C1 − 16 )x2 + ( 13 C12 − 16 C2 )x − 24 C22 = 0 for the second equation of Example 6.12. 2 has the canonical form An equation in symmetry class S3,4 0

v 00 − ae−v = 0

(7.28)

326 with constant a 6= 0. Two integrations lead immediately to the general solution  au + C1  log (au + C1 ) − 1 + C2 . (7.29) v= a 2 2 Similar to S3,1 and S3,2 there is a single constant that has to be determined from the canonical form of the given equation. It is substituted together with the transformation functions into (7.29) in order to generate the solution for the given equation. The algorithm below performs these steps, it returns the solution for any equation in this symmetry class.

Algorithm 7.10 LieSolve2.3.4(ω). Given a second order quasilinear or2 dinary differential equation in symmetry class S3,4 , the general solution is returned. S1 : Janet basis for symmetries. Generate a Janet basis for the symmetry generators of ω. S2 : System for transformation functions. If the Janet basis obtained in (2,2) S1 has type J3,6 , set up system (6.29), (6.30) or (6.31) respectively. If (2,2)

it has type J3,7

set up system (6.32).

S3 : Determine transformation functions. Determine two independent Liouvillian solutions for the system obtained in S2. S4 : Return solution. Generate the canonical form and determine the value for a from it. Substitute a and the canonical variables into (7.29) and return the result. Example 7.13 The transformation functions for an equation in symmetry 2 class S3,4 have been obtained in Example 6.13 as σ = x2 and ρ = x + y 2 . p √ √ Substituting the inverse x = u, y = v − u leads to a canonical form (7.28) with a = 1. Substituting a and the transformation functions into (7.29) yields finally the solution   y 2 = (C1 + x2 ) log (C1 + x2 ) − 1 − x + C2 for the equation considered in Example 5.16 Eight-Parameter Projective Symmetry. The canonical form v 00 (u) = 0 with the general solution v = C1 + C2 u of an equation with this symmetry is particularly simple because it does not contain any unspecified elements. Solving such an equation therefore comes down to determining the transformation functions σ and ρ. According to Theorem 6.14 this is a two-step procedure. At first a rational solution of the partial Riccati system (6.37) has to be found. Secondly, an independent set of Liouvillian solutions σ and ρ has to be determined by solving (6.36). The various steps of this proceeding are put together in the subsequent algorithm.

Solution Algorithms

327

Algorithm 7.11 LieSolve2.8(ω). Given a quasilinear second order ode ω in symmetry class S82 . Its general solution is returned; or failed if it cannot be determined. S1 : Determine a and b. From the coefficients of ω generate the system (6.37) and determine a special rational solution. If none is found return failed. S2 : System for transformation functions. Apply the result of S1 and construct the Janet basis (6.36). S3 : Determine transformation functions. Find two independent Liouvillian solutions of the Janet basis constructed in S2. If none are found return failed. S4 : Return solution. Substitute u = σ(x, y) and v = ρ(x, y) into the canonical form v = C1 + C2 u and return the result. Example 7.14 Equation 6.180 has been considered already in Example 6.14. Substituting the values for u ≡ σ(x, y) and v ≡ ρ(x, y) obtained there into v = C1 + C2 u yields after some simplifications the general solution y=

C2 x2 − x . C1 + C2 x2 − x

Example 7.15 Equation 6.124 has been considered before in Example 6.15. Substituting the values for u ≡ σ(x, y) and v ≡ ρ(x, y) obtained there into v = C1 + C2 u yields after some simplifications the general solution 1 y=p . C1 e−x + C2 e−2x Lie’s Second Integration Method. In his book on differential equations Lie [113], Kapitel 20, §4 and 5, a solution scheme for differential equations with symmetries is described that he called second method of integration. It is based on the observation that any second order quasilinear ode y 00 = ω(x, y, y 0 ) dy 00 dy 0 may be written as the system y 0 = ,y = which in turn is equivalent dx dx to the linear pde ∂f ∂f ∂f + y0 + ω(x, y, y 0 ) 0 = 0. ∂x ∂y ∂y The application of this second integration scheme is based on the following lemma. Lemma 7.2 (Lie 1891) The differential equation y 00 + r(x, y, y 0 ) = 0 allows the symmetry with the generator U = ξ∂x + η∂y if and only if the linear pde Af ≡

∂f ∂f ∂f + y0 − r(x, y, y 0 ) 0 = 0 ∂x ∂y ∂y

allows the first extension of U , i.e., if [U, A] = ρ(x, y, y 0 )A for some function ρ.

328 The proof may be found in the above quoted book by Lie. Based on this lemma, the problem of solving an ode with symmetries is replaced by a linear pde allowing the once extended vector fields and the theory of Chapter 2, Section 5 applies. This scheme is applied in the following example. Example 7.16 Equation 6.159 has been solved in Example 7.9. For this 3 y 02 , its symmetry generators are (see Example 5.23) equation r = −3y 2 − 4y U1 = ∂x and U2 = x∂x − 2y∂y . Therefore the system ∂x f + y 0 ∂y f + 3y 2 −

3 02
y ∂y0 f = 0, ∂x f = 0, x∂x f − 2y∂y f − 3y 0 ∂y0 f = 0 4y

is obtained. It leads to ∆ = 32 y 02 − 6y 3 and the integral (2.47) yields Z  Z √ 0 y y 3 02  dy 2 0 dy Φ1 = . 3y + y − y = log 02 4y ∆ ∆ y − 4y 3 3 02 Its exponential is the first integral u02 v obtained in Example 7.9 written v −1 in actual variables x and y.

7.3

Nonlinear Equations of Third Order

In this section a complete discussion of symmetry methods for solving genuinely nonlinear equations of third order is presented, similar to second order equations above. In Lie’s book [113] on differential equations there is only a short chapter on third order equations allowing certain three-parameter symmetry groups. One-Parameter Symmetry. A quasilinear third order ode in symmetry class S13 has the canonical form v 000 + r(v 00 , v 0 , u) = 0.

(7.30)

Substituting v 0 = w(u) yields w00 + r(w0 , w, u) = 0.

(7.31)

This is a general second order equation for w(u) because r(w0 , w, u) is an undetermined function of its arguments as it has been explained in the introduction on page 311. If it has a symmetry, the results of the preceding section may be applied for solving it, if not it is returned by the following algorithm performing these steps.

Solution Algorithms

329

Algorithm 7.12 LieSolve3.1(ω). Given a quasilinear third order ode ω in symmetry class S13 , its general solution or a lower order equation is returned, or failed if neither may be found. S1 : Janet basis for symmetries. Generate a Janet basis for the determining system of the Lie symmetries of ω. S2 : Determine transformation functions. If case a) of Theorem 6.7 applies choose σ. If case b) applies solve the first order equation for σ. If it cannot be obtained return failed. Determine ρ by integration. S3 : Canonical form. Applying the solutions obtained in S2 generate the canonical form (7.31). If it does not allow any nontrivial symmetry return it. S4 : Return solution. Proceed with the proper solution procedure for a second order equation. If a solution is found, resubstitute the actual variables and return the result. Example 7.17 The canonical form obtained in Example 5.27 leads to the second order equation (7.30) w00 w − 3w02 − (u − 4)w0 w2 + (2u − 4)w4 + w3 = 0. Because it does not have any nontrivial symmetry, this is the best possible answer in the context of Lie’s theory. Two-Parameter Symmetries. According to the discussion in the introduction of this chapter, a third order equation with a two-parameter symmetry in general may be reduced to a first order ode of type (7.1). For the four twoparameter symmetry classes discussed there these are the equations (7.32), (7.35), (7.39) and (7.42) below. The first symmetry class to be discussed is 3 S2,1 . It has the canonical form v 000 + r(v 00 , v 0 ) = 0.

(7.32)

00

Substituting v 000 = v 00 dv 0 yields dv dv 00 r(v 00 , v 0 ) + = 0. dv 0 v 00

(7.33)

This is a general first order equation for v 00 (v 0 ) because r(v 00 , v 0 ) is an undetermined function of its arguments. If v 00 = w(v 0 , C1 ) is its general solution, an integration yields Z dv 0 = u + C2 . (7.34) w(v 0 , C1 ) If the integration may be performed, the resulting first order ode allows the one-parameter symmetry ∂v . Therefore the desired solution of (7.32) may be represented in terms of an integral.

330 The various steps are put together in the algorithm below. The solution procedure may fail in step S2 if the transformation functions are not Liouvillian, or in step S4 if equation (7.33) cannot be solved. Algorithm 7.13 LieSolve3.2.1(ω). Given a quasilinear third order ode 3 ω in symmetry class S2,1 , its general solution or some lower order equation is returned. S1 : Equations for σ and ρ. Generate a Janet basis for the symmetry generators of ω and set up system (6.3). S2 : Determine transformation functions. Determine an independent pair of Liouvillian solutions of the system obtained in step S1. If it cannot be found return failed. S3 : Second order equation. By means of the transformation functions obtained in S2 generate equation (7.33) and determine its solution. If it cannot be found return (7.33). S4 : Integral representation. Perform the integration at the left hand side of (7.34). If it cannot be done return (7.34). S5 : Return solution. Solve the first order equation obtained in step S4 and return the result. If a solution cannot be found return the first order equation and the transformation functions. Example 7.18 The canonical equation v 000 +2v 00 v 0 +1 = 0 of Example 6.17 has the form (7.32). With v 00 = w and v 0 = z the Abel equation of second kind ww0 + 2wz + 1 = 0 for w(z) is obtained. According to Chapter 4.2 it may be transformed into an Abel equation of first kind for which a symmetry cannot be found. 3 A quasilinear third order ode in symmetry class S2,2 has the form

v 000 +

1 r(uv 00 , v 0 ) = 0. u2

(7.35)

Substituting first v 0 = z yields the equation z 00 + 12 r(uz 0 , z) = 0 allowing the u one-parameter group u∂u . By Theorem 6.7 new variables w ¯ and t defined by u = ew¯ , z = t with w ¯ ≡ w(t), ¯ and finally w ¯ 0 (t) = w(t) yield the first order equation  1  w0 + w2 − w3 r , t = 0. (7.36) w If its general solution is w = w0 (t, C1 ), the relation Z  u = C2 exp w0 (v 0 , C1 )dv 0 (7.37) between v 0 and u involving two constants follows. If the integration may be performed, the solution procedure may be continued. The subsequent algorithm performs these steps. The crucial step is S3. If it fails the first order equation (7.36) is the best possible answer.

Solution Algorithms

331

Algorithm 7.14 LieSolve3.2.2(ω). Given a quasilinear third order ode 3 ω in symmetry class S2,2 , its general solution or some lower order equation is returned. S1 : Equations for σ and ρ. Generate a Janet basis for the symmetry generators of ω. S2 : Determine transformation functions. Determine the transformation functions σ from (6.6), and ρ from (6.7) or (6.8). S3 : First order equation. By means of the transformation functions obtained in S2 generate equation (7.36) and determine its solution. If it cannot be found return (7.36). S4 : Integral representation. Perform the integration at the left hand side of (7.34). If it cannot be done return (7.37). S5 : Return solution. Solve the first order equation obtained in step S4 and return the result. If a solution cannot be found return the first order equation and the transformation functions. It may occur that the above second order equation for z allows a larger symmetry group than just the one-parameter generator u∂u as it is shown in the example below. In these cases it is advantageous to apply this larger symmetry for the further solution procedure. Example 7.19 The canonical equation obtained in Example 6.18 has the form (7.35). It turns out that after substituting v 0 = z the resulting equation 5 z(z 2 + 4) 3 =0 z 00 − z 02 − z 0 − z u u2

(7.38)

belongs to symmetry class S82 . Consequently, algorithm LieSolve2.8 may be 4C1 applied. It yields the solution z = q , and finally 4 C2 u + u2 − 4C12 Z x=

dy q

+ C3

C1 y 2 + C2 − 41 y 4

in actual variables. In Exercise 7.5 the same result is obtained from the first order equation (7.36). 3 A quasilinear third order ode in symmetry class S2,3 has the form

v 000 + v 0 r

 v 00  , u = 0. v0

Substituting first v 0 = z yields z 00 + zr

 z0   z0  z 00 , u = 0 or +r , u = 0. z z z

(7.39)

332 0 Finally substituting zz = w leads to

w0 + w2 + r(w, u) = 0.

(7.40)

If this general first order equation may be solved, and w = w0 (u, C1 ) is its general solution, the solution of (7.39) is Z Z  v = C2 exp w0 (u, C1 )du du + C3 . (7.41) The subsequent algorithm may terminate in step S2 without generating the desired solution if the first order ode for σ cannot be solved, or in step S3 if equation (7.40) cannot be solved. Algorithm 7.15 LieSolve3.2.3(ω). Given a quasilinear third order ode 3 ω in symmetry class S2,3 , its general solution or some lower order equation is returned. S1 : Equations for σ and ρ. Generate a Janet basis for the symmetry generators of ω, and set up equation (6.41) for σ and equations (6.39),..., (6.43) for ρ. S2 : Determine transformation functions. Choose σ or solve (6.41). If σ is not known at this point return failed. Otherwise determine independent solution for ρ. S3 : Second order equation. By means of the transformation functions obtained in S2 generate equation (7.40) and determine its solution. If it cannot be found return (7.40). S4 : Integral representation. Perform the integration at the left hand side of (7.41). If it cannot be done return (7.41). S5 : Return solution. Solve the first order equation obtained in step S4 and return the result. If a solution cannot be found return the first order equation and the transformation functions. Example 7.20 The canonical equation obtained in Example 6.19 has the form (7.39) from which the Abel equation (7.40) w0 + w3 + w2 + u + 1 = 0 follows. Because a symmetry cannot be found for it, this first order equation is the best possible answer in the realm of symmetry analysis. 3 A quasilinear third order ode in symmetry class S2,4 has the form

v 000 + r(v 00 , u) = 0.

(7.42)

Substituting v 00 = w the general first order equation w0 + r(w, u) = 0

(7.43)

is obtained. If w = w0 (u, C1 ) is its general solution, the solution of (7.42) is Z Z v= w0 (u, C1 )du du + C2 u + C3 . (7.44)

Solution Algorithms

333

The subsequent algorithm applies these steps. It may fail in step S2, or a first order equation may be returned in step S3 as the best possible answer of the symmetry analysis. Algorithm 7.16 LieSolve3.2.4(ω). Given a quasilinear third order ode 3 ω in symmetry class S2,4 , its general solution or some lower order equation is returned. S1 : Equations for R or S. Generate a Janet basis for the symmetry generators of ω and set up equations (6.44), (6.45) or (6.46). S2 : Determine R or S. From the equations obtained in S1 determine rational solution for R or S. If none is found return failed. S2 : Determine transformation functions. Applying the solution obtained in S2 determine σ and ρ by integration. S4 : First order equation. Generate the first order equation (7.43) with the help of σ and ρ. If a solution cannot be found return this equation. S5 : Integral representation. Using the solution obtained in S4 return the integral representation (7.44) and the transformation functions. Example 7.21 The canonical equation obtained in Example 6.20 has the form (7.42) from which the Riccati equation (7.43) w0 + uw2 + w + 1 = 0 follows. Because a closed form solution cannot be found it is the best possible result of the symmetry analysis. Three-Parameter Symmetries. The general solution procedure for the equations covered in this subsection has been described in Theorem 7.1 in the introduction to this section. According to the classification of symmetries given in Theorem 5.13 and the listing of groups and Lie algebras on page 135, the first two symmetry classes with nonsolvable Lie algebras of symmetry generators require solving a Riccati equation, i.e., equations (7.46) and (7.49) below. In the remaining cases with a solvable symmetry algebra the solution reduces essentially to integrations. 3 has a canonical form The first symmetry class S3,1  v 00 (u − v) + 2v 0 (v 0 + 1)  6v 00 (v 0 + 1) 6v 0 (v 02 + 4v 0 + 1) v 02 √ + + r = 0. u−v (u − v)2 (u − v)2 v0 v0 (7.45) 0 0 √ v 00 (u − v) + 2v (v + 1) 0 √ Introducing w = v and z = as new variables, the v0 v0 Riccati equation w2 − 12 zw + 1 dw + =0 (7.46) dz r(z) − 32 z 2 − 12

v 000 +

for w ≡ w(z) is obtained. If its general solution may be determined in the form Φ(v 0 , z) = C1 , two more independent relations between u, v, v 0 and z may be obtained by differentiation, see Lie [109], page 288.   (v − u)v 0 Φv0 = C2 and (v − u)v 0 (v − u)v 0 Φv0 v0 + 2vΦv0 = C3 . (7.47)

334 From these relations v may be obtained by elimination. These steps are applied in the subsequent algorithm. It may terminate in step S2 if a rational solution of the respective Riccati equations may not be found, or in step S4 if equation (7.46) cannot be solved. Algorithm 7.17 LieSolve3.3.1(ω). Given a quasilinear third order ode ω 3 in symmetry class S3,1 , its general solution is returned; or failed if it cannot be determined. S1 : Equations for R and S. Generate Janet basis for the symmetry generators of u and set up system (6.13), (6.16) or (6.18) for R, or system (6.20) for R and S respectively. S2 : Determine R and S. From the equations obtained in S1 determine a rational solution for R and S. If none can be found return failed. S3 : Determine transformation functions. Applying the result of S2 set up the system (6.14), (6.17), (6.19) or (6.21) for σ and ρ respectively and determine two independent Liouvillian solutions. S4 : Riccati equation. Applying the transformation functions obtained in step S3 generate equation (7.46). If its solution cannot be determined return this equation. S5 : Return solution. Set up system (7.47), eliminate v, resubstitute the actual variables and return the result. Example 7.22 The canonical equation obtained in Example 6.21 has the form (7.45) with r(z) = z 2 + 1 and z as given above. It leads to the 2 Riccati equation (7.49) w0 − 2w 2− zw + 2 = 0 for w(z). Because a closed z + 22 form solution cannot be found it is the best possible result of the symmetry analysis. 3 A quasilinear third order ode in symmetry class S3,2 has the canonical form

v 000 v 5 + 3v 00 v 0 v 4 + r(v 00 v 3 ) = 0.

(7.48)

If the two lowest invariants w ≡ vv 0 and z ≡ v 00 v 3 of the two-parameter subgroup {∂u , 2u∂u + v∂v } are introduced as new dependent and independent variables and dw = vv 00 + v 02 , dz = v 000 v 3 + 3v 00 v 0 v 2 are applied, (7.48) du du assumes the form dw 1 2 z + w + = 0. (7.49) dz r(z) r(z) This Riccati equation for w(z) has to be solved. If w = w0 (z, C) is its general solution, in the original variables u and v a second order equation of the form w0 (v 00 v 3 , C) − vv 0 = 0 is obtained allowing the above mentioned two-parameter symmetry group. Because it is not in the canonical form as applied in Theorem 5.8, this second

Solution Algorithms

335

order ode is not in canonical ¯ and √ form (7.17). Introducing new variables u ¯ yields ∂v¯ and u ¯∂u¯ + v¯∂v¯ , and the second order v¯(¯ u) by u = v¯ − u ¯ and v = u equation   2¯ 1 uv¯00 + v¯0 − 1 , C − w0 − = 0. (7.50) 0 3 0 4(¯ v − 1) 2(¯ v − 1) Under suitable constraints for the function w0 it may be treated by the algorithm LieSolve2.2.2 on page 322. These steps are applied in the algorithm below. It may terminate in step S2 if a rational solution for R cannot be found, or in step S5 if (7.49) cannot be solved, or if (7.50) is too complicated. Algorithm 7.18 LieSolve3.3.2(ω). Given a quasilinear third order ode ω 3 in symmetry class S3,2 , its general solution is returned; or failed if it cannot be found. S1 : Equations for R. Generate Janet basis for the symmetry generators of ω. Set up system (6.22), (6.23) or (6.24) for R from it. S2 : Determine R. From the equations obtained in S1 determine a rational solution for R. If it cannot be found return failed. S3 : System for transformation functions. Applying the result of S2 set up the systems (6.22), (6.23) or (6.24) for σ and ρ. S4 : Determine transformation functions. Determine two independent Liouvillian solutions for the system obtained in S3. S5 : Riccati equation. Generate equation (7.49) and determine a rational solution. If it cannot be found return this equation. S6 : Return solution. Employing the solution obtained in S5, generate (7.50), apply algorithm LieSolve2.2.2 to it and return the result. Example 7.23 The canonical equation obtained in Example 6.22 has the form (7.48) with r(v 00 v 3 ) = 2v 00 v 3 + 1 and leads to the Riccati equation (7.49) dw + 1 w2 + z 2z + 1 = 0. Because a closed form solution cannot be found dz 2z + 1 it is the best possible result of the symmetry analysis. 3 has the form A quasilinear third order ode in symmetry class S3,3
v 000 + v 0(c−3)/(c−1) r v 00 v 0−(c−2)/(c−1) = 0.

(7.51)

00 With v 000 = dv 0 v 00 it may be rewritten as a first order ode for v 00 (v 0 ) dv


dv 00 00 v + v 0(c−3)/(c−1) r v 00 v 0−(c−2)/(c−1) = 0 0 dv with the symmetry (c − 1)v 0 ∂v0 + (c − 2)v 00 ∂v00 . Applying Theorem 6.1, new variables u ¯ and v¯(¯ u) are introduced by v 0 = u ¯−(c−1)/(c−2) e(c−1)¯v and 00 (c−2)¯ v v =e with the result Z c−1 1 d¯ u v¯ = + C1 ≡ R(¯ u) + C1 . (7.52) 2 c−2 (c − 1)r(¯ u) + (c − 2)¯ u u ¯

336 If the original variables are resubstituted by u ¯ = v 00 v 0−(c−2)/(c−1) and 1 00 v¯ = c − 2 log v , the equation R



v 00 v 0(c−2)/(c−1)



−

1 log v 00 + C1 = 0 c−2

(7.53)

is obtained. If R does not involve integrals, this is a second order ode allowing the two symmetries ∂u and ∂v . Under suitable constraints for R it may be treated by the algorithm LieSolve2.2.1 on page 322. The subsequent algorithm is designed according to this scheme. The only bottleneck where it may fail is the integral (7.52). Algorithm 7.19 LieSolve3.3.3(ω). Given a third order quasilinear or3 dinary differential equation in symmetry class S3,3 (c), its general solution is returned, or equation (7.53) if it cannot be determined. S1 : Janet basis for symmetries. Generate a Janet basis for the symmetry generators of ω and set up system (6.25), (6.26), (6.27) or (6.28) respectively. S2 : Determine transformation functions. Determine two independent Liouvillian solutions for the system obtained in S2. S3 : Integral representation. Generate (7.53). If the integration may not be executed in closed form return this integral. 2 S4 : Return solution. If (7.53) is in S2,2 , apply algorithm LieSolve2.2.1 to it and return the result. Otherwise return (7.53). Example 7.24 If into the canonical form v 000 v 03 + 1 = 0 of Example 6.23 a new dependent variable is introduced by v 0 = w, the second order equation 2 w00 w3 +1 = 0 in symmetry class S3,2 is obtained. Applying (7.25) an additional integration yields its solution p v = 12 (u + C2 ) C12 (u + C2 )2 − 1 1 log C (u + C ) + pC 2 (u + C )2 − 1 + C . − 2C 1 2 2 3 1 1 Substituting the canonical variables determined in Example 6.23, the general solution 1 (C x + 1)pC 2 (C x + 1)2 − x2 + 1 log x y = 2x 2 2 1 2C1 p 1 − 2C log C1 (C2 x + 1) + C12 (C2 x + 1)2 − x2 + C3 1 of the equation in Example 5.34 is finally obtained. 3 A quasilinear third order ode in symmetry class S3,4 has the form 0

v 000 + e−2v r v 00 ev

0




= 0.

(7.54)

Solution Algorithms

337

00 Writing v 000 = dv 0 v 00 yields the first order equation dv 0

0 e−2v dv 00 + r(v 00 ev ) = 0 0 00 v dv

(7.55)

for v 00 (v 0 ) with the symmetry ∂v0 − v 00 ∂v00 . Applying Theorem 6.1, new variables u ¯ and v¯(¯ u) are introduced by v 0 = log u ¯ + v¯ and v 00 = e−¯v with the result Z u r(¯ u) d¯ + C1 ≡ R(¯ u) + C1 . (7.56) v¯ = 2 u ¯ − r(¯ u) u ¯ 0

Upon resubstitution of the original variables by u ¯ = v 00 ev and v¯ = − log v 00 , the equation 0
R v 00 ev + log v 00 + C1 = 0 (7.57) is obtained. If R does not involve integrals, this is a second order ode allowing the two symmetries ∂u and ∂v . Similar to the preceding case, the only bottleneck where the algorithm below may fail is the integral (7.56). Algorithm 7.20 LieSolve3.3.4(ω). Given a third order quasilinear or3 dinary differential equation ω in symmetry class S3,4 , the general solution is returned, or equation (7.56) if it cannot be determined. S1 : Janet basis for symmetries. Generate a Janet basis for the symmetry generators of ω and set up system (6.29), (6.30), (6.31) or (6.32) respectively. S2 : Determine transformation functions. Determine two independent Liouvillian solutions for the system obtained in S2. S3 : Integral representation. Generate (7.57). If the integration may not be performed return this integral. S4 : Return solution. Apply algorithm LieSolve2.2.1 to the equation obtained in S3 and return the result. Example 7.25 The canonical equation obtained in Example 6.24 has the 0 0 form √(7.54) r(v 00 ev ) = v 002 e2v + 1. In step S2 the equation √ 0 with −v 0 00 v − 2 v − C1 e = 0 is obtained. Finally, R step S4 yields the parameter representation u = f (p) + C and v = f (p)dp − pf (p) + C3 where 2 Z p e dp p f (p) = − √1 . 2 p − C1 3 has the form A quasilinear third order ode in symmetry class S3,5

v 000 + v 0 r

 v 00  = 0. v0

(7.58)

00 Substituting v 0 = z yields the first order equation dz + z 2 + r(z) = 0 from du v which Z dz + u = C1 (7.59) 2 z + r(z)

338 follows. If the integration may be performed and the result may be solved for z, i.e., z = z0 (u, C1 ), the second order linear equation v 00 − z0 (u, C1 )v 0 = 0 is obtained with the general solution Z Z  v = C2 exp z0 (u, C1 )du du + C3 .

(7.60)

(7.61)

The algorithm below is based on this solution scheme. It may fail in step S3 if the integration in (7.59) may not be executed, or if the result of the integration may not be solved for z. In these cases an expression involving v 00 and v 0 is returned. On the other hand, if equation (7.60) may be obtained, the general solution is returned. Algorithm 7.21 LieSolve3.3.5(ω). Given a quasilinear third order ode ω 3 in symmetry class S3,5 , its general solution is returned. S1 : Janet basis for symmetries. Generate Janet basis for symmetry generators of ω and set up equations (6.48), (6.49), (6.50), (6.51) or (6.52) respectively. S2 : Determine transformation functions. From the equations obtained in S2 determine two independent Liouvillian solutions. S3 : Integral representation. Applying the solutions obtained in S3 generate the equation (7.59). If z0 (u, C1 ) cannot be obtained, return this expression. S4 : Return solution. Generate equation (7.60) and its solution (7.61), resubstitute actual variables and return the result. 3 The symmetry class S3,5 is one of the few cases of third order equations considered by Lie [113], pages 556-558. He uses canonical variables u ¯ = v and v¯ = u such that the group generators have the form ∂u¯ , ∂v¯ and u ¯∂u¯ 000 00 v¯00 = 0. with invariants v¯02 , v¯03 and the canonical equation v 000 + v¯00 r v¯ v¯ v¯03 Its integration leads finally to the same answer (7.61). Of particular interest are Lie’s remarks on page 556, line 7 to 10 from the bottom, concerning the comparison of this proceeding with his second integration method described on page 327.1

The canonical form obtained in Example 6.25 corresponds 4u to r(z) = z − 2 and the solution z = Ce4u + 1 . Substituting it into the Ce − 1 second order equation v 00 − zv 0 = 0 yields Z h


i 1 2 v = C1 +C2 u + log C3 eu − 1 +log C3 eu + 1 +log C32 e2u + 1 du. 2 Example 7.26 2

1 Translation

the above... .

by the author: In certain cases this method requires simpler operations than

Solution Algorithms

339

3 3 3 The four symmetry classes S3,6 to S3,8 and S3,9 (c) arise from the groups g17 in Chapter 3, page 139. Their common canonical form

dz + r(z) = 0 where z(u) ≡ v 00 + c1 v 0 + c2 v du

(7.62)

is linear and homogeneous in v ≡ v(u), v 0 and v 00 , with constant coefficients c1 and c2 . This first order equation for z may always be integrated with the result Z dz + u + C ≡ R(z) + u + C = 0. (7.63) r(z) This relation determines z either explicitly in the form z = z0 (u, C) or implicitly as function of u. In any case v is the solution of the linear equation v 00 + c1 v 0 + c2 v = z0 (u, C).

(7.64)

The left hand side of this equation is always completely reducible (see Theorem 6.16). Therefore its general solution may be written as follows. RR 3 S3,6 : v = C1 + C2 u + z0 (u, C)dudu, R 3 S3,7 : v = C1 eu + C2 ueu − (u + 1)e−u z0 (u, C) + (u + 1)e−u z0 (u, C)du, R R 3 S3,8 : v = C1 + C2 eu + z0 (u, C) + z0 (u, C)du − eu e−u z00 (u, C)du, u R cu R 3 S3,9 (c) : v = C1 eu + C2 ecu + ec e−u z0 (u, C)du − ec e−cu z0 (u, C)du. (7.65) Resubstituting the actual variables into these expressions finally yields the solution of the given equation. Due to the similiar course of the solution procedure for all four groups under consideration, the various steps have been put together in the following algorithm. It may fail in step S3 if a hyperexponential solution for ρ cannot be found.

Algorithm 7.22 LieSolve3.3.6to9(ω). Given a quasilinear third order ode 3 3 3 to S3,8 or S3,9 (c), its general solution ω in one of the symmetry classes S3,6 is returned; or failed if it cannot be determined. S1 : Equations for transformation functions. Generate Janet basis for symmetry generators of ω and set up equations for σ and ρ as described in Theorems 6.20, 6.21, 6.22 or 6.23 respectively. S2 : Determine transformation functions. From the equations obtained in S1 determine a pair of independent Liouvillian solutions for σ and hyperexponential solutions for ρ. If the latter may not be found return failed. S3 : Canonical form. Applying the solutions obtained in S2 generate the canonical form (7.64).

340 S4 : Return solution. Substitute the value of z0 and the actual variables into the appropriate expression (7.65) and return the result. Example 7.27 The canonical form equation dz + r(z) = 0 with r(z) = dx 2 − z z+ 1 that has been obtained in Example 7.27 yields R(z) = 12 log z 2 + 1 √ and, consequently, z0 (u, C) = Ce−2u − 1. This value for z0 (u, C) has to be substituted into the expressions for v given above. Finally resubstituting 1 and v = y and renaming C into C yields the the actual variables by u = x 3 solution Z Z 1 dx  dx z¯0 (x, C3 ) 2 y = C1 + C2 + x x x2 q of the first ode’s in Example 5.37 with z¯0 (x, C3 ) = C3 exp (− x2 ) − 1. Proceeding along the same lines, the answer for the remaining three ode’s is v = C1 exp

 1   1    1  1 1  + C2 exp − 1+ exp − z¯0 (x, C3 ) x x x x x Z   1   dx 1 z¯0 (x, C3 + 1 + exp − , x x x2

Z  1  dx v = C1 + C2 exp + z¯0 (x, C3 ) − z¯0 (x, C3 ) 2 x x  1   1 Z exp − z¯0 (x, C3 )dx, + exp x x 0 Z  c  1h  1  1 1 dx v = C1 exp + C1 exp + exp ( ) exp (− )¯ z0 (x, C3 ) 2 x x c x x x Z  c  c dx i − exp ( ) exp − z¯0 (x, C3 ) 2 x x x respectively. 3 A quasilinear third order ode in symmetry class S3,10 has the canonical form v 000 + v 002 r(v 0 ) = 0. (7.66) 00 00 Substituting v 000 = v 00 dv 0 yields the linear first order equation dv 0 + r(v 0 )v 00 dv dv = 0 for v 00 (v 0 ). Integration leads to  Z  v 00 + C1 exp − r(v 0 )dv 0 = 0. (7.67)

If the integral may be performed, this is a second order ode in symmetry class 2 S2,1 . For special forms of r(v 0 ) its solution may be obtained explicitly. In

Solution Algorithms any case, defining R(p) = parameter representation

R

341

R exp ( r(p)dp)dp the expressions (7.16) yield the Z

u + C1 R(p) = C2 , v + C1 pR(p) = C1

R(p)dp + C3 .

(7.68)

The following algorithm is designed according to this scheme. For any equation in this symmetry class at least the parameter representation (7.68) is returned. Algorithm 7.23 LieSolve3.3.10(ω). Given a quasilinear third order ode 3 ω in symmetry class S3,10 , its general solution is returned either explicitly or as a parameter representation S1 : Equations for transformation functions. Generate Janet basis for symmetry generators of ω and set up equations (6.56) for σ and ρ. S2 : Determine transformation functions. From the equations obtained in S1 determine a pair of independent Liouvillian solutions or failed if it may not be found. S3 : Canonical form. Applying the solutions obtained in S3 generate the equation (7.67). S4 : Return solution. Apply algorithm LieSolve2.2.1 to the equation obtained in S3 and return the result. Example 7.28 The canonical form of the ode considered in Example 6.27 3 in symmetry class S3,10 corresponds to r(v 0 ) = v 0 + 10 . For this particular v choice of r the integrals at the left hand side of (7.67) may be executed in closed form with the result v 02 = 2 log (C1 u + C2 ). It leads to the general solution √ Z p v= 2 log (C1 u + C2 )du + C3 . 1 and v = y yields the solution Substituting u = x r √ Z 1 1 y=− 2 log (C1 + C2 ) 2 dx + C3 x x of the ode in Example 5.38. Four-Parameter Symmetries. In the remaining part of this section the number of symmetries exceeds the order of the given equation at least by one. Consequently, the solution procedure is simplified. It requires only algebraic operations. 3 The first symmetry class S4,1 has a canonical form v 000 v 0 + av 002 = 0

(7.69)

with constant a 6= 0, a 6= − 32 . Dividing by v 00 v 0 and performing three consecutive integrations leads to the general solution (a + 2)a+1 (C1 v + C2 )a+1 = (a + 1)a+2 (C1 u + C3 )a+2 .

342 In this form the invariance under exchange of u and v combined with the replacement of a by −(a + 3) is obvious. Usually the explicit form v = (C1 u + C2 )(a+2)/(a+1) + C3

(7.70)

with new constants Ck is more convenient for solving equations. The constant a together with the transformation functions has to be substituted into (7.70) in order to generate the solution for the given equation. The subsequent algorithm applies these steps for obtaining the general solution. For any equation in this symmetry class it is guaranteed that the solution is found. Algorithm 7.24 LieSolve3.4.1(ω). Given a quasilinear third order ode ω 3 in symmetry class S4,1 , its general solution is returned. S1 : Equations for transformation functions. Generate Janet basis for symmetry generators of ω and set up equations (6.57), (6.58), (6.59) or (6.60) respectively depending on the type of the Janet basis. S2 : Determine transformation functions. From the equations obtained in S1 generate an independent pair of solutions. S3 : Canonical form. Applying the solutions obtained in S2 generate the canonical form (7.69). S4 : Return solution. Substitute the value of a and the actual variables into (7.70) and return the result. Example 7.29 Substituting the values a = − 21 and u = xy, v = y1 into (7.70) ensures the solution (C1 xy + C2 )3 y + C3 y + 12C1 = 0, i.e., a third order algebraic function. It is easy to see that a = − 25 and u = y1 , v = xy lead to the same answer. 3 The canonical form of an ode in symmetry class S4,2 is

√ v 000 v + 3v 00 v 0 + av 00 v 00 v = 0.

(7.71)

Its symmetry group contains {∂u , 2u∂u + v∂v , u2 ∂u + uv∂v } as a subgroup. 3 Writing (7.71) as v 000 v 5 + 3v 00 v 0 v 4 + a(v 00 v 3 ) 2 = 0 shows that it is a special 3 case of (7.48) with r(v 00 v) = a(v 00 v 3 ) 2 . This special form of r is a consequence of the additional symmetry v∂v . Applying the same transformation which led to (7.49) yields a special Riccati equation with an additional symmetry. It is 0 more efficient however to introduce the new dependent variable w = vv which has to satisfy w00 + 6ww0 + 4w3 + a(w0 + w2 )3/2 = 0. Its general solution is rational and leads to the general solution Z a2 (x + C2 )dx v = C1 exp a2 x(x + C3 ) + 4C22 − (a2 − 16)C2 (C2 − C3 )

(7.72)

Solution Algorithms

343

of (7.71). Another method for arriving at this result is discussed in Exercise 7.11. The subsequent algorithm implements the various steps. The desired solution is only returned if rational transformation functions in step S2 may be found. Algorithm 7.25 LieSolve3.4.2(ω). Given a quasilinear third order ode ω 3 in symmetry class S4,2 , its general solution is returned; or failed if it cannot be determined. S1 : Equations for R. Generate Janet basis for symmetry generators of ω and set up equations for R according to (6.61) or (6.62) depending on the type of the Janet basis. S2 : Determine R. From the equations obtained in S1 determine rational solution for R. If none may be found return failed. S3 : Equations for transformation functions. Applying the solutions obtained in S2 set up equations (6.61) or (6.61) for σ and ρ. S4 : Return solution. Applying the transformation function obtained in S3 generate (7.72), resubstitute the actual variables and return the result. 3 For an ode in symmetry class S4,3 (c) it is assumed that the value of the parameter c is already known. The canonical form of an equation in this symmetry class is

v 000 + av 00α = 0 with α =

c−3 . c−2

(7.73)

Substituting v 00 = w leads to w0 + awα = 0. Three successive integrations yield the general solution 1 (α − 1)(3−2α)/(1−α) (au + C1 )(3−2α)/(1−α) + C2 u + C3 . a (2 − α)(3 − 2α) (7.74) The subsequent algorithm implements the various steps. The desired solution is only returned if hyperexponential transformation functions in step S2 may be found. v=

2

Algorithm 7.26 LieSolve3.4.3(ω). Given a quasilinear third order ode ω 3 in symmetry class S4,3 (c), its general solution is returned; or failed if it cannot be determined. S1 : Equations for transformation functions. Generate Janet basis for the symmetry generators of ω and set up equations (6.63), (6.64) or (6.65) respectively depending on the type of Janet basis. S2 : Determine transformation functions. From the equations obtained in S1 generate an independent pair of hyperexponential solutions. If none is found return failed. S3 : Canonical form. Applying the solutions obtained in S3 generate the canonical form (7.73).

344 S4 : Return solution. Substitute the value of a and the actual variables into (7.74) and return the result. Example 7.30 Substituting the values a = 1, c = 1 , v = xy into (7.74) ensures the general solution u= x √ 1
3/2 + C2 + C3 . y = 23x2 C1 + x x x2

3 2,

i.e., α = 3 and

3 Finally the canonical form of an ode in symmetry class S4,4 is


v 000 + a exp − 12 v 00 = 0.

(7.75)

Substituting v 00 = w leads to w0 + a exp (− 21 w) = 0. Similar to the preceding case, three succesive integrations lead to
v = (u − C1 )2 log(u − C1 ) + log (− 12 a) − 32 u2 + C2 u + C3 . (7.76) The subsequent algorithm implements the various steps. The desired solution is only returned if hyperexponential transformation functions in step S2 may be found. Algorithm 7.27 LieSolve3.4.4(ω). Given a quasilinear third order ode ω 3 in symmetry class S4,4 , its general solution is returned; or failed if it cannot be determined. S1 : Equations for transformation functions. Generate a Janet basis for the symmetry generators of ω and set up equations (6.66), (6.67), (6.68) or (6.69). S2 : Determine transformation functions. From the equations obtained in S1 generate an independent pair of hyperexponential solutions. If none is found return failed. S3 : Canonical form. Applying the solutions obtained in S3 generate the canonical form (7.75). S4 : Return solution. Substitute the value of a and the actual variables into (7.76) and return the result. Six-Parameter Symmetry S63 . In canonical variables an ode in symmetry class S63 has the form v 000 v 0 − 32 v 002 = 0. (7.77) The subgroup {∂u , ∂v , u∂u , v∂v } may be utilized for its integration. The special value a = − 32 in (7.70) leads to the general solution v=

C1 + C3 u + C2

after a suitable change of the integration constants. The algorithm below applies these steps. It may fail if rational solutions for R and S cannot be found.

Solution Algorithms

345

Algorithm 7.28 LieSolve3.6(ω). Given a quasilinear third order ode ω in symmetry class S63 , its general solution is returned; or failed if it cannot be determined. S1 : Equations for R and S. If u has the form (5.40) set up the equations from (6.70) and (6.71). If ω has the form (5.41), set up equations from (6.72) and (6.73). S2 : Determine R and S. From the equations obtained in S1 determine a rational solution for R and S. If none may be found return failed. S3 : System for transformation functions. Applying the result of S2 set up the system from (6.70), (6.71) or (6.72) and (6.73) for σ and ρ. S4 : Determine transformation functions. Determine two independent Liouvillian solutions for the system obtained in S3. S5 : Return solution. Substitute the solutions from S4 into (7.77) and return the result. Example 7.31 For the equation with symmetry type S63 and structure y (5.40) in Example 5.41 the transformation functions σ = xy and ρ = x have been obtained. Substituting these values into (7.77) leads to the general solution x3 y 2 + (C1 x2 + C2 )y + C3 x = 0. Example 7.32 For the equation in symmetry class S63 and structure (5.41) y of Example 5.42 the transformation functions σ = y1 and ρ = x ex have been obtained. Substituting these values into (7.77) leads to the general solution C1 y 2 + (C2 xe−x + 1)y + C3 xe−x = 0.

7.4

Linearizable Third Order Equations

The equations considered in this section are assumed to be equivalent to a linear ode. Consequently, the methods described in the first section of Chapter 2 may be applied after the canonical form has been obtained. Four-Parameter Symmetry. The canonical form for any equation in the 3 only symmetry class S4,5 of this subsection is the generic linear third order equation in rational normal form v 000 + P (u)v 0 + Q(u)v = 0.

(7.78)

The coefficients P (u) and Q(u) may assume quite different shapes by transformations of the structure invariance group u = F (¯ u) and v = F 0 (¯ u)¯ v . The further proceeding depends on the structure of the differential Galois group of (7.78) which may allow it to obtain solutions explicitly, e. g. if it is reducible with a nontrivial Loewy decomposition. In general, however, it is irreducible

346 and the only information that is guaranteed for equations in this symmetry class is its linearizability. Algorithm 7.29 LieSolve3.45(ω). Given a quasilinear third order ode ω 3 in symmetry class S4,5 , its general solution is returned; or failed if it cannot be determined. S1 : Identify case a), b) or c). Determine the Janet basis for the Lie symmetries of ω. By means of the criteria given in Theorem 6.31 identify the case that applies for equation ω. S2 : Equations for σ and ρ. From the Janet basis obtained in step S1 construct the system for ρ according to (6.74), (6.76) or (6.78) respectively. In case c) set up the first order ode for σ. S3 : Determine σ and ρ. In case a) and b) choose σ, in case c) determine it from the first order ode. From the equations constructed in step S2 find a hyperexponential solution for ρ, if none exists return failed. S4 : Canonical form equation. Applying the transformation functions from step S3 generate a canonical form equation, determine its Loewy decomposition and construct the most general explicit solution from it. S5 : Find solution. Substitute u = σ(x, y) and v = ρ(x, y) into the canonical form solution obtained in step S4 and return the result. Example 7.33 The first canonical form (6.80) obtained in Example 6.32
1
0 0 has the Loewy decomposition v 0 + 2u4u − 1 v Lclm v − 2v, v − u v = 0 from which the general solution 2u

v = C1 u + C2 e

+ C3

h

2u

e

Z

ue−4u du − u (2u − 1)2

Z

i e−2u du (2u − 1)2

follows. Substituting the transformation u = x and v = x y given there yields the solution of the equation in Example 5.43 in the form y= C1 x + C2

e2x

+ C3

h

e2x

Z

x Z i. e−2x xe−4x dx − x dx (2x − 1)2 (2x − 1)2

(7.79)

The second canonical form (6.81) has the Loewy decomposition h

v0 +



  1 1 2  i 1 2 2 + − 2 v Lclm v 0 − v, v 0 − ( − 2 )v = 0 u−2 u u u u u

from which the solution v = C1 u + C2 u2 exp

Z Z 4 )du 2 )du i h exp (− u exp (− u 2 2 + C3 u2 exp − u u u u(u − 2)2 (u − 2)2

1 and v = x yields again (7.79). is obtained. Substituting u = x y

Solution Algorithms

347

Example 7.34 The equation considered in Example 5.44 generates the (2,2) type J4,19 Janet basis, therefore case b) applies. If σ = y is chosen, the system (6.74) yields the independent solution ρ = x y and the canonical form 1 1 (6.80). Choosing σ = y instead yields ρ = xy with the canonical form (6.81). Example 7.35 The equation considered in Example 5.45 generates the (2,2) type J4,17 Janet basis, therefore case c) applies. The first order equation for 0 σ is y + 1 = 0, i.e., φ(x, y) = x + y. Choosing σ = x + y yields a system for ρ with the fundamental system {0, x, y}. Consequently, any linear combination ρ = c1 x + c2 y with c1 6= c2 yields a possible transformation to a canonical form which turns out to be (6.80). Five-Parameter Symmetries. The canonical form for the symmetry classes 3 3 S5,1 and S5,2 (a) is v 000 −(a+1)v 00 +av 0 = 0 with a some number. This canonical form is particularly convenient because it is completely reducible for any value of a without introducing new algebraic numbers. If a = 0 it decomposes as v 000 − v 00 = Lclm v 0 , v 0 −


1 v, v 0 − v = 0 u

(7.80)

with the general solution v = C1 + C2 u + C3 eu , and if a 6= 0 v 000 − (a + 1)v 00 + av 0 = Lclm(v 0 , v 0 − v, v 0 − av) = 0

(7.81)

with the general solution v = C1 + C2 eu + C3 eau . Substituting ρ and σ for v and u immediately yields the desired solution of the actual equation. These 3 steps are organized in terms of the subsequent algorithm for both S5,1 and 3 S5,2 (c) symmetries. Algorithm 7.30 LieSolve3.5(ω). Given a quasilinear third order ode ω with any five-parameter group of Lie symmetries, its general solution is returned; or failed if it cannot be determined. S1 : Identify case a), b) or c). Determine the Janet basis for the Lie symmetries of ω. By means of the criteria given in Theorem 6.11, identify the case that applies for the equation at issue. S2 : Determine value of parameter a. Solve equation (5.42) and choose some value of a from it. S3 : Equations for σ and ρ. From the Janet basis constructed in step S1 and the value for a obtained in step S2 construct the systems for σ and ρ according to (6.82) and (6.83), (6.84) and (6.32) or (6.85) and (6.86). S4 : Determine σ and ρ. From the equations constructed in step S3 determine σ by integration. Find a hyperexponential solution for ρ, if none exists return failed. S5 : Return solution. Substitute u = σ(x, y) and v = ρ(x, y) into the canonical form solution (7.80) or (7.81) and return the result.

348 y Example 7.36 The transformation functions σ = xy and ρ = x obtained in Example 6.35 yield the general solution (C3 x2 + 1)y + C2 exy x + C1 x = 0 for the differential equation considered in Example 5.46. Example 7.37 The transformation functions σ = y and ρ = x y obtained in Example 6.36 yield the general solution
C3 e3x + C2 ey + C1 y + x = 0 for the differential equation considered in Example 5.47. Seven-Parameter Symmetry. In canonical variables u and v ≡ v(u), a quasilinear third order ode in symmetry class S73 has the form v 000 = 0 with the obvious solution v = C1 +C2 u+C3 u2 . Therefore, the whole solution procedure comes down to generating the canonical form of the given equation, very much like for a second order equation with eight-parameter projective symmetry group. For third order equations the two alternatives of Theorems 5.18 and 6.33 have to be distinguished. The various steps are put together in the subsequent algorithm. Algorithm 7.31 LieSolve3.7(ω). Given a quasilinear third order ode ω with a seven-parameter group of Lie symmetries, its general solution is returned; or failed if it cannot be determined. S1 : Identify case a) or case b). By means of the criteria given in Theorem 5.18 determine which case applies for equation ω. S2 : Equations for σ and ρ. Determine a solution for S from equations (6.87) or (6.87). If none is found return failed; otherwise construct the system for σ from it. S3 : Determine σ and ρ. From the equations constructed in step S2 determine Liouvillian transformation functions σ and ρ. S4 : Return solution. Substitute u = σ(x, y) and v = ρ(x, y) into the canonical form v = C1 + C2 u + C2 u2 and return the result. Example 7.38 For equation 7.8 of Kamke’s collection in Example 6.37 the transformation functions σ = x and ρ = √2 − 2 have been obtained. y Substituting them into the above solution for v 000 = 0 yields the solution 4 √2 − 2 = C1 + C2 x + C3 x2 or equivalently y y=

1 (C1 + C2 x + C3 )2

after a redefinition of the integration constants. Example 7.39 For the equation with symmetry type S73 and structure y (5.44) in Example 5.49 the transformation functions σ = xy and ρ = x have

Solution Algorithms

349

been obtained. Substituting these values into the above solution for v 000 = 0 leads to the general solution x3 y 2 + (C1 x2 + C2 )y + C3 )x = 0. Example 7.40 For the equation with symmetry type S73 and structure y (5.44) in Example 5.50 the transformation functions σ = y1 and ρ = x ex have been obtained. Substituting these values into the above solution for v 000 = 0 leads to the general solution y 3 + (C1 y 2 + C2 y + C3 )xe−x = 0.

Exercises Exercise 7.1

Apply Lemma 7.1 for solving the equation in Example 6.2.

Exercise 7.2 Solve the equation of Example 7.2 by transforming it into the form v 0 = C. Exercise 7.3 Solve the equation of Example 7.2 under the assumption that only the second symmetry generator U2 = y∂x + x∂y is known. Exercise 7.4

Derive the expression (7.16) for the solution of (7.13).

Exercise 7.5 Generate the solution of the equation considered in Example 7.19 from the first order equation (7.36). Exercise 7.6 Determine the solution of equation Kamke 6.227 from the canonical form obtained in Example 6.8. Exercise 7.7

Solve the equation y 00 + r(y) = 0 by symmetry methods.

Exercise 7.8

Solve the equation y 00 y + r(x)y 02 = 0 by symmetry methods.

Exercise 7.9 Determine the solution of (7.24) from the equivalent pde and compare the result with (7.25). Exercise 7.10 Derive a first order Riccati equation from (7.71) and determine its solution. Exercise 7.11

Derive solution (7.72) from the equivalent pde of (7.71).

Chapter 8 Concluding Remarks

After discussing the various integration methods for ode’s developed by Lie it appears appropriate to conclude with a few words of comparison, assess its advantages and limitations, and also put them into proper perspective. Furthermore, several directions of future research are outlined. Comparison of Lie’s Methods. There are basically two alternative methods for solving differential equations using its symmetries. Either an equation with a certain nontrivial symmetry is transformed into a canonical form the solution of which may be determined. Or the infinitesimal symmetry generators are determined explicitly, and are used to determine a set of first integrals from which the solution may be constructed. The relation between these two alternatives is most easily understood from the following drawing. The order n of the given quasilinear ode is two or three. Equation in actual variables y (n) + r(x, y, y, . . . , y (n−1) ) = 0 1h ? Generate Janet basis for determining system, determine symmetry type  HH  2h  H 5h  HH  j H  h Generators U1 , . . . , Ur Equation in canonical variables 6a  Ui = ξi (x, y)∂x + ηi (x, y)∂y v (n) + s(u, v, v, . . . , v (n−1) ) = 0 3h

6h

? ? Integrals from equivalent pde Solution in canonical variables HH  h  4hHH 7 H  j H   Solution in actual variables Step 1 takes the given ode in actual variables as input, constructs the determining system from it and transforms it into a Janet basis. To this end

351

352 only differentiations and arithmetic in the base field are required. From the Janet basis coefficients the symmetry type may be obtained as described in Theorems 5.9 and 5.14. This step is guaranteed and it is always performed first. After that there are two main alternatives. The left branch which is emphasized in this book begins with step 2. At first a system of equations for the transformation functions to canonical form is set up. If an independent pair of solutions for the transformation functions may be obtained, it is applied for generating the canonical form equation for the respective symmetry class. In step 3 its solution, or at least a simplified equation, is determined. In the final step 4 this result is transformed back to the actual variables x and y. Alternatively the vector fields generating the symmetries may be determined explicitly in step 5. They are applied in step 6 to determine a set of first integrals for the pde which is equivalent to the given ode. In step 7 by elimination the desired solution or some lower order equation may be constructed from the first integrals. These two main branches are not completely disconnected. One may leave the right branch with step 6a and then proceed along the left branch. There are several reasons why the left branch in the above drawing is preferred. i) The equations which determine the transformations to canonical form are usually simpler than the equations determining the symmetry generators. For all possible types of equations occurring in the symmetry analysis of second and third order ode’s algorithms have been described for decomposing them into components of lowest order. ii) If the symmetry generators are determined explicitly, they have to be transformed such that they obey the canonical commutators of the respective Lie algebra. iii) The first integrals for the equivalent pde often lead to unnecessary complicated expressions for the solution of the given ode which are difficult to simplify. There is an additional, more fundamental reason. In general, solving a differential equation may be considered as an equivalence problem consisting of two parts. At first membership of the given equation to a particular equivalence class has to be decided. Secondly, within this equivalence class the transformation of the given equation to a canonical form has to be determined algorithmically. This scheme is valid independent of whether an equation has symmetries or not. In the former case, the symmetry class of the equation narrows down the possible equivalence classes, and the canonical form for it is already known. These aspects are clearly discussed by Tresse [181] as it is explained in Section 4.3 on page 188. If there are no symmetries, candidates for equivalence classes are usually specified in terms of differential equations defining certain special functions

Concluding Remarks

353

like e. g. Painlev´e transcendents, Bessel functions, etc. As soon as membership to any of them is established, the transformation to the canonical equation has to be found. These two steps may be combined into a single one. This approach is dicussed in the Dissertation of Berth [10]. Problems for Further Research. It would be highly desirable to admit more general base fields, e. g. elementary extensions by one or more exponentials, angular functions or logarithms, or algebraic functions. Because the base field occurs as coefficient field for the Janet basis of the determining system, and also for the equations determining the transformation to canonical form, the decompositions of these Janet bases over more general coefficient fields must be known first if an algorithmic theory without heuristics is the goal. For various applications it might be of interest to have a similar scheme available for equations of order four comprising more than one hundred symmetry classes. Lie himself announced a book on point and contact symmetries of partial differential equations. Due to his early decease he has not been able to realize this project; and so another field of possible research is left. Due to the large number of possible symmetry types that have to be considered, a complete classification is rather voluminous and relies even more on appropriate computer algebra tools. Similar arguments apply to the symmetry analysis of systems of ode’s. There have been various attempts to extend Lie’s symmetry concept in order to make more equations amenable to a solution procedure. To this end, Barbara Abraham-Shrauner et al. [1] applied nonlocal symmetries. Muriel et al. [134, 49] define the new concept of a potential symmetry. A completely different approach has been pursued by Zaitsev and Polyanin [191]. As opposed to Lie, they apply discrete groups for solving ordinary differential equations. It would be highly interesting to compare their results with those obtained by Lie analysis. Another comparison concerns the relation between Galois’ theory and Lie’s symmetry analysis. For second order linear equations there is certainly no chance for any correlation because any such equation has the full eightparameter projective group as its symmetry group. The situation is different, however, for equations of order three as may be seen from the listings in Appendix D and E. For example, all equations in symmetry class S73 have 3 a Loewy decomposition of type L39 . All equations in symmetry class S5,2 (c) 3 3 have Loewy decompositions of type L9 or type L12 . This suggests that for linear equations of order higher than two there may be some correlations between its symmetry type and the type of its Loewy decomposition, which in turn entails a special Galois group.

Appendix A Solutions to Selected Problems

In this Appendix the solutions to most exercises are given, although in some cases only a few hints are provided. The reader is strongly advised to try first on his own to find the answers.

2. Linear Differential Equations 2.1 The constraints for the coefficients of the given system of algebraic equations are a1 = 2b4 a3 b3 − b2 a3 + b33 − b23 a4 + b3 a2 + c3 a3 , b1 = b2 b3 − a3 b24 − c4 a3 − b4 b23 , c1 = a2 c4 + a2 b24 − b22 + b2 c3 − b3 c2 + a3 b34 + b4 c4 a3 + b24 b23 , c2 = a4 c4 − 2b2 b4 + a4 b24 + b4 c3 + b24 b3 − c4 b3 . 2.2 If y1 = z12 is differentiated three times, and z100 + 41 pz1 = 0 is applied for reduction whenever possible, one obtains y10 = 2z1 z10 , y100 = − 12 pz12 + 2z102 and y1000 = − 12 p0 z12 − 2pz1 z10 . Substitution into the given third order ode shows that y1 is a solution. Similarly differentiating and substituting y2 = z1 z2 , y20 = z10 z2 + z1 z20 , y200 = − 12 pz1 z2 + 2z10 z20 and y2000 = − 21 p0 z12 − 2pz1 z10 shows that y2 is a solution as well. The same is true for y3 . 2.3 Let y1 , y2 and y3 be a fundamental system for the given equation and W its Wronskian. The general solution may be written as Z

r (y2 y30 − y20 y3 )dx W

Z

r (y1 y30 − y10 y3 )dx + W

y = C1 y1 + C2 y2 + C3 y3 + −

Z

r (y1 y20 − y10 y2 )dx W

where C1 , C2 and C3 are constants. For y 000 = r with the fundamental system y1 = 1, y2 = x and y3 = x2 there follows R R R y = C1 + C2 x + C3 x2 + 21 x2 rdx − x xrdx + 21 x2 rdx. 2.4 Reducing y 00 + P y 0 + Qy = 0 w.r.t. y 0 + py = 0 and y 0 + qy = 0 yields Q − pP = p0 − p2 and Q − qP = q 0 − q 2 . Solving for P and Q yields R the given expressions. Substituting the fundamental system y1 = exp (− pdx),

355

356 R R y2 = exp (− qdx) and the Wronskian W = (q − p) exp (− (p + q)dx) into (2.9) yields  Z R R + C y = exp (− pdx) exp ( pdx) qRdx 1 −p Z  R R + exp (− qdx) exp ( qdx) qRdx − p + C2 . This representation generalizes the solution of a first order equation in an obvious way. 2.5 For L22 , the first solution y1 is obtained from (D + a1 )y = 0, the second from (D + a1 )y = y¯2 with y¯2 a solution of (D + a2 )y = 0. For L23 the two solutions are obtained from (D + ai )y = 0. Linear dependence over the base field would imply a relation q1 y1 + q2 y2 = 0 with q q1 , q2 from the base field. Substituting the solutions this would entail q21 = R − exp (a2 − a1 )dx. Due to the nonequivalence of a1 and a2 , its difference is not a logarithmic derivative. Consequently the right hand side cannot be rational. For L24 the equation (D + a(C))y = 0 has to be solved; then C is specialized 0 to C1 and C2 . Substituting ai = r +r C + p in the above quotient, the i q C2 integration may be performed with the result q12 = − rr + + C1 which is rational. 2.6 Let y 000 + ay 00 + by 0 + cy = 0 with fundamental system y1 , y2 , y3 be the third order linear equation corresponding to the second order Riccati equation y0 z 00 + 3zz 0 + z 3 + a(z 0 + z 2 ) + bz + c = 0. If yii is rational for a single value of i, or for two or three values of i, but for any pair i, j with 1 ≤ i < j ≤ 3, the ratios Ci yi0 + Cj yj0 (A.1) Ci yi + Cj yj are not, the corresponding number of rational solutions of the Riccati equation exists. This is case i) for the second order equation. If there is a pair of indices such that the ratio (2.17) is rational, case ii) or iii) respectively applies depending on whether the logarithmic derivative of the remaining element of the fundamental system is rational or not. Finally, if the ratio C1 y10 + C2 y20 + C3 y30 C1 y1 + C2 y2 + C3 y3 is rational, it may be rewritten as Cy10 /y1 + C2 y20 /y1 + y30 /y1 C2 (y2 /y1 )0 + (y3 /y1 )0 y0 = + 1 C1 + C2 y2 /y1 + y3 /y1 C1 + C2 (y2 /y1 ) + y3 /y1 y1

(A.2)

0 0 C1 and C2 appropriately redefined and p = y1 , q = y2 , with the ratios C y1 y1 C3 3 y30 y2 y3 r = y1 , u = y1 and v = y1 . This is case iv).

Solutions to Selected Problems

357

2.7 If the Galois group is Z1 , by Theorem 2.7 the equation has a rational fundamental system {y1 , y2 }; i.e., it is linearly dependent over the base field. By the result of Exercise 2.4 the Loewy decomposition is of type L24 . If the group is Z2 , its second symmetric power has a rational fundamental system y2 y y12 , y1 y2 and y22 . Consequently, the ratio y1 y1 2 = y12 is rational and the decomposition L24 follows as above. 2.8 Substituting P (u) ≡ u2 + B1 u + b0 = 0 into u0 + u2 + r = 0 leads to b0 = r − 12 (b01 − b21 ) and b1 b01 − 2b01 + 12 b1 (b21 − 4b0 ). If the expression for b0 is substituted into this latter relation, the constraint b001 − 3b1 b01 + b31 + 4rb1 − 2r0 = 0 0 follows. With b1 = − zz the third order equation z 000 + 4rz 0 + 2r0 z = 0 for z is obtained which is the the second symmetric power of y 00 + ry = 0. 2.9 Substituting z1 and z2 into the equations of the given Janet basis leads to a linear system for the coefficients a1 , a2 , b1 and b2 . Defining

z z z z z z w0 = 1,x 2,x , w1 = 1,y 2,y , w2 = 1,y 2,y , z1,x z2,x z 1 z2 z1 z 2 z z z z w3 = 1,xx 2,xx , w4 = 1,yy 2,yy , z1 z2 z 1 z2 z1,yy z2,yy z1,xx z2,xx , w6 = w5 = z1,x z2,x z1,y z2,y they may be expressed as a1 = −

w1 w2 w3 w5 , a2 = , b1 = − , b2 = . w0 w0 w0 w0

2.10 With the same notation as in the previous exercise the representations z z w6 4 a1 = − z1,x = − z2,x , b1 = − w w1 and b2 = w1 are obtained. 1 2 2.11 By direct reduction the following Riccati-like system of pde’s is obtained. a1,x + A1 a21 + (B1 − A2 )a1 + a2 − B2 = 0, a1,y + B1 a21 − a1 a2 + (C1 − B2 )a1 − C2 = 0, a2,x + A1 a1 a2 + B1 a2 − A3 a1 − B3 = 0, a2,y −

a22

+ B1 a1 a2 + C1 a2 − B3 a1 − C3 = 0,

b1 + A1 a1 − A2 = 0, b2 + A1 a2 − A3 = 0.

(A.3)

358 2.12 Substituting (2.79) into system (2.81) and reordering yields after a few autoreduction steps the following result.  A1,x − A − 2B w0,x 2 1 A   1 A1,x A − A + 2A B − 2A B w0 = 0, − 2 2,x 1 2 2 1 A1 w0,xy + A2 w0,y − A1 (B1,x − A1 B2 + A2 B1 − A3 − B12 )w0,x 1 h i A2 (B − B 2 + A2 B − A A ) w = 0, + B2,x + A2 B2 − B1 B2 + B3 − A 1,x 1 2 3 0 1 2 w0,xx − A1 w0,y −



1

w0,yy + (2B2 + C1 )w0,y − A1 (C1,x − A1 C2 + 2B1 B2 − 2B3 )w0,x 1 i h 2 (C 2B B − 2B ) w0 = 0. + C2,x + 2B22 + 2C3 − A + 1 2 3 1,x A 1

2.13 Each term in (2.19) is uniquely determined by an index sequence k0 , k1 , . . . , kν−1 . If the lexicographic ordering with respect to this sequence is assumed, the leading terms corresponding to these sequences are given in the subsequent table. Furthermore, the sum of the indices is given. N o. k0

k1 k2 k3 . . . kν−2 kν−1

ν−1 X

kj (M + j)

j=0

1

0 0 0 ...

0

0

νM

2 ν − 2 1 0 0 ...

0

0

(ν − 1)M + 1

3 ν − 3 0 1 0 ...

0

0

(ν − 2)M + 2

4 ν − 4 2 0 0 ...

0

0

(ν − 2)M + 2

5 ν − 4 0 0 1 ...

0

0

(ν − 3)M + 3

0

.. . 1

.. . M +ν−1

.. . last

ν

.. . 0

... 0 0 0 ...

The rightmost column contains the order of a pole of order M for the term determined by the values in the preceding columns. Because ν ≥ 1, the value νM in the first line is maximal if M > 1. For M = 1 they are all H equal to ν. 2.14 The first member of a fundamental system is z1 = exp (− adx + bdy). Dividing out w1 ≡ zx + az and w2 ≡ zy + bz yields the system w2 + A1 w1 = 0, w1,x + (B1 − a)w1 = 0, w1,y + (A1,x + A1 + A2 − A1 B1 )w1 = 0 with the solution (see Lemma 2.9)  I  w ¯1 = exp − (B1 − a)dx + (A1,x + A1 + A2 − A1 B1 )dy , w ¯2 = −A1 w ¯1 .

Solutions to Selected Problems

359

The corresponding inhomogeneous system zx + az = w ¯1 , zy + bz = −A1 w ¯1 finally yields the second member of a fundamental system (see Lemma 2.10) I 1 z2 = z1 w z1 (dx − A1 dy). H 2.15 The first member of a fundamental system is z1,1 = exp (− bdx + cdy), z2,1 = −az1,1 . Dividing out the Loewy factor w1 ≡ z2 + az1 , w2 ≡ z1,x + bz1 and w3 ≡ z1,y + cz1 yields the system w1,x + (aA1 + C1 )w1 = 0, w1,y + (aB1 + D1 )w1 = 0, w2 + A1 w1 = 0, w3 + B1 w1 = 0 with the solution (see Lemma 2.9)  I  w ¯1 = exp − (aA1 + C1 )dx + (aB1 + D1 )dy , w ¯2 = −A1 w ¯1 , w ¯3 = −B1 w ¯1 . The corresponding inhomogeneous system z2 + az1 = w ¯1 , z1,x + bz1 = −A1 w ¯1 , z1,y + cz1 = −B1 w ¯1 finally yields the second member of a fundamental system (see Lemma 2.10) I w ¯1 (A1 dx + B1 dy), z2,2 = −az1,2 . z1,2 = −z1,1 z1,1 2.16 Reduction w.r.t. the given two first order systems yields a linear algebraic system for the coeffients of the desired Lclm. If a1 6= a2 , the result is (1,2) of type J2,1 , − b2 z + a1 b2 − a2 b1 z = 0, zy − ab1 − a2 x a1 − a2 1 a − a2,x a1 a2,x − a2 a1,x zxx + (a1 + a2 − 1,x )z = 0. a1 − a2 )zx + (a1 a2 + a1 − a2 (1,2)

If a1 = a2 and b1 6= b2 , the result is of type J2,1 ,   b1,y − b2,y  b1 b2,y − b1,y b2  zx + a1 z = 0, zyy + b1 + b2 − zy + b 1 b 2 − z = 0. b 1 − b2 b 1 − b2 2.17 Introducing a differential indeterminate z, the system of linear pde’s zy −

x zx = 0, y

zxx −

xy − 1 y zx − z = 0 x x

(A.4)

is obtained. The right factor leads to zx − yz = 0, zy − xz = 0 with the solution exy . It is the first basis element for the solution space of (A.4). Introducing coordinates z1 and z2 in D2 , the exact quotient generates the 1 1 pde’s z2 − x y z1 = 0, z1,x + x z1 = 0 and z1,y = 0 with the solution z1 = x ,

360 z2 = y1 . In order to obtain the second basis element for the solution space 1 , z − xz = 1 has to be of (A.4), the inhomogeneous system zx − yz = x y y xy solved with the result e Ei(−xy). The exponential integral is defined by Z ∞ dt e−t . Ei(−x) = − t x 2.18 A special solution of the inhomogeneous system is searched for in the form z = C1 z1 + C2 z2 where both C1 ≡ C1 (x, y) and C2 ≡ C2 (x, y) are undetermined functions of x and y. They will be obtained by an extension of Lagrange’s variation of constants. The last equation yields z1 , z2 wz2 wz1 . C1,x = − , C2,x = where W = z1,x , z2,x W W Substituting the expression for z into the first equation and the equation zxy + (a1,x + a2 − a1 b1 )zx + (a2,x − a1 b2 )z = vx − a1 w that is obtained from it by derivation w.r.t. y yields C1,y z1 + C2,y z2 = v, C1,y z1,x + C2,y z2,x = vx − a1 w. By elimination the expressions C1,y =

vz1,x − (vx − a1 w)z1 vz2,x − (vx − a1 w)z2 , C2,y = − W W

for the y-derivatives are obtained. Combining these results, both coefficients C1 and C2 , and consequently a special solution of the inhomogeneous system, may be obtained as a path integral. 2.19 A special solution of the inhomogeneous system is searched for in the form z = C1 z1 + C2 z2 with z , z2 vz2,y − vy z2 vy z1 − vz1,y , C1,x = , C2,x = where W = 1 z1,y , z2,y W W wz1 wz2 , C2,y = . W W It is most easily obtained from the preceding case by exchange of x and y and substituting a = 0. 2.20 A special solution of the inhomogeneous system is searched for in the form z = C1 z1 + C2 z2 + C3 z3 , Ck ≡ Ck (x, y) for k = 1, 2, 3 are undetermined functions of x and y. Lagrange’s variation of constants leads to v, z2 , z3 w z2 , z3 1 vx , z2,x , z3,x , C1,x = , C1,y = W z2,x , z3,x W vxx − aw, z2,xx , z3,xx C1,y = −

Solutions to Selected Problems

361

z1 , v, z3 1 w z , z3 z1,x , vx , z3,x , , C2,y = C2,x = − 1 W z1,x , z3,x W z1,xx , vxx − aw, z3,xx z , z2 , v w z1 , z2 1 1 z , z , v C3,x = , C = 1,x 2,x x 3,y W z1,x , z2,x W z1,xx , z2,xx , vxx − aw where z1 , z 2 , z 3 W = z1,x , z2,x , z3,x . z1,xx , z2,xx , z3,xx From these expressions the Ck (x, y) may be expressed in terms of path integrals. 2.21 With the substitutions xk ≡ φk (y1 , . . . , yn ), yk ≡ ψ(x1 , . . . , xn ) and g(y1 , . . . , yn ) ≡ f (φ1 , . . . , φn ), the partial derivatives of f become ∂f ∂g ∂y1 ∂g ∂yn = + ... + . ∂xi ∂y1 ∂xi ∂yn ∂xi Substituting them into (2.45), rearranging terms and defining Bk = a1

∂ψk ∂ψk + . . . + an ∂x1 ∂xn

for k = 1, . . . , n yields (2.46). 2.22 With the notation of Theorem 2.20, the equation for w(x, y, z) is dy 1 wy + awx + rwz = 0. From dy = dx a there follows dx = a and the first integral ψ1 (x, y). Introducing a new variable x ¯ = ψ1 (x, y) with the inverse x = φ(¯ x, y), the equation for w becomes wy + (b(φ, y)z + c(φ, y))wz = 0. From dy = dz there follows dz − b(φ, y)z = c(φ, y). It yields a second first bz + c dy integral h  Z  Z  Z  i ψ2 (x, y, z) = z exp − b(φ, y)dy − c(φ, y) exp − b(φ, y)dy dy

x ¯=ψ1

The general integral of the originally given equation may be written as Φ(ψ1 , ψ2 ) = C with a constant C. The particular choice of the function Φ determines the actual form of the answer. As an example, consider the y2 1 z equation zy + 2x 3y zx = 3y z with the first integrals ψ1 = x3 and ψ2 = y 1/3 . ψ2 y Choosing Φ = 1/3 yields z = C x . ψ1 2.24 Taking the commutator with A of both sides of the assumed relations

.

362 yields k (A, Bk ) = (A, βm+1 Bm+1 ) + . . . + (A, βqk Bq ) + (A, αA) k · Bm+1 + . . . + Aβqk · Bq = Aβm+1 k +βm+1 (A, Bm+1 ) + . . . + βqk (A, Bq ) + Aα · A.

By assumption the commutators involving the Bk in the last line are proportional to A; i.e., the complete expression is a relation of the same type given in the Theorem which is a contradiction. Consequently, Aβjk = 0 for all k and j. 2.25 The second equation has the general solution y xy ¯ . − z(x, y, C(x)) = ¯ x C(x) + y x2 y − 2 Substitution into (2.55) yields C¯x x + C¯ = 0; i.e., C¯ = C x with C a constant y xy follows. from which the general rational solution z = xy + C − 2 x y−2 1 into (2.50) yields the linear system 2.26 Substituting z = z1 + w wx − (2A1 z1 + A2 )w = A1 , wy − (2B1 z1 + B2 )w = B1 for w in complete analogy to a first order ordinary Riccati equation. 2.27 If in the system zx + A1 z 2 + A2 z + A3 = 0, zy + B2 z + B3 = 0 the new w , the following system for w is obtained. function w is introduced by z = A 1

wx + w2 +



A2 −

 A1,x  A1,y  w + A1 A3 = 0, wy + B2 − w + A1 B3 = 0. A1 A1

The coefficient of w in the latter equation vanishes identically due to the first constraint in (2.51). The second constraint of (2.51) yields for the last term A1 B3 =

1 A1,x  A2 − . 2 A1 y

The y-integration may be performed with the result w=

 1  A1,x − A2 +C 2 A1

where C ≡ C(x) is a function of x. Substituting this expression for w into the first equation yields the Riccati equation (2.57) for the x-dependence of C. If the terms not containing C are derived w.r.t. y and the constraints (2.51) are applied for eliminating all y-derivatives, zero is obtained. If the solution w is used, the for C is substituted into the above expression for w and z = A 1 representation (2.56) for the solution follows.

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2.28 Dividing out the exact component {zx ≡ z1 , zy ≡ z2 } yields {z2 + az1 , z1,x + cz1 , z1,y + (ax − ac)z1 }. The corresponding linear system has the general solution R z1 = exp (− cdx + (ax − ac)dy), z2 = −az1 . It leads to the system zx = z1 , zy = −az1 with the solution R R R z = z1 dx − az1 dy = exp (− cdx + (ax − ac)dy)(dx − ady) which is the non-constant member of a fundamental system.

3. Lie Transformation Groups 3.1 The differential equations ∂x ¯ = 1, ∂a

∂x ¯ = 0, ∂b

∂ y¯ = 0, ∂a

∂ y¯ =1 ∂b

have the required form; however, there does not exist an identity element. 3.2 Let ak correspond to the one-parameter group Uk . Then the finite transformations are x ¯=

ea3 y (x + a1 )ea2 , y¯ = a2 1 − (x + a1 )e a4 1 − (x + a1 )ea2 a4

3.3 The finite transformations of g2 are x ¯ = a1 x + a2 y + a3 , y¯ = a4 x + a5 y + a6 . Accordingly, three arbitrarily chosen points (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ) are transformed into x¯i = a1 xi + a2 yi + a3 and y¯i = a4 xi + a5 yi + a6 for i = 1, 2, 3. These relations may be solved for the group parameters as follows. x x1 x x1 y1 1 ¯1 y1 1 ¯1 1 1 1 1 x ¯2 y2 1 , a2 = x2 x ¯2 1 , a3 = x2 y2 1 , a1 = D D D x ¯ 3 y3 1 x3 x ¯3 1 x ¯1 x ¯2 x ¯3 y¯1 y1 1 x1 y¯1 1 x1 y1 1 1 1 1 y¯2 y2 1 , a5 = x2 y¯2 1 , a6 = x2 y2 1 , a4 = D D D y¯3 y3 1 x3 y¯3 1 y¯1 y¯2 y¯3 x1 y1 1 D is defined by x2 y2 1 . x3 y3 1 F (x) y ≡ σ(x) and v = ≡ ρ(x, y), 3.4 Introducing new variables u = 2 F1 (x) F1 (x) F (x) Lemma 3.3 yields vanishing coefficients for all ∂u and Gk (u) = 2 with x F1 (x) expressed by u for k = 1, . . . , r. Obviously G1 = 1 and G2 (u) = u as required.

364 3.5 Applying the same notation as in Example 3.1, the following system of equations for the undetermined coefficients is obtained. α12 α23 − α13 α22 = α11 , α11 α23 − α13 α21 = −α12 , α12 α33 − α13 α32 = 2α21 = 0, α11 α33 − α13 α31 = −2α22 = 0, α22 α33 − α23 α32 = α31 = 0, α21 α33 − α23 α31 = −α32 = 0 and α13 = α23 = α33 = 0. Obviously, it allows only the trivial solution αij = 0 for all i and j.
3.6 For r = 2 there holds ∆(ω) = ω 2 − c212 E1 − c112 E2 ω, for r = 3   ∆(ω) = ω 3 − (c212 + c313 )E1 − (c112 − c323 )E2 − (c113 + c223 )E3 ω 2  + (c212 c313 − c213 c312 )E12 − (c112 c323 − c123 c312 )E22 + (c113 c223 − c123 c213 )E32 −(c112 c313 − c113 c312 − c212 c323 + c223 c312 )E1 E2 +(c112 c213 − c113 c212 + c213 c323 − c223 c313 )E1 E3  +(c112 c223 − c113 c323 − c123 c212 + c123 c313 )E2 E3 ω. 3.7 For l = 1, k = 1, ρ1 ≥ 1. For any value of ρ1 , there are two groups of size ρ1 + 2 corresponding to α1 = 0 or α1 = 1, respectively. Their generators and non-vanishing commutators are ∂y , x∂y , . . . , xρ1 ∂y , ∂x with [∂x , xk ∂y ] = kxk−1 ∂y and ex ∂y , xex ∂y , . . . , xρ1 ex ∂y , ∂x

with [ex ∂x , xk ∂y ] = kxk−1 ex ∂y + xk ex ∂y .

For l = 2, 1 ≤ k ≤ 2, the constraint ρ1 + ρ2 ≥ 0 becomes redundant. For any pair of values ρ1 and ρ2 , there are two groups of size ρ1 + ρ2 + 3 corresponding to α1 = 0 or α1 = 1 respectively. Their generators and nonvanishing commutators are ∂y , x∂y , . . . , xρ1 ∂y , eα2 x ∂y , xeα2 x ∂y , . . . , xρ2 eα2 x ∂y , ∂x with [∂x , xk ∂y ] = kxk−1 ∂y ,

[∂x , xk eα2 x ∂y ] = kxk−1 eα2 x ∂y + α2 xk eα2 x ∂y

and ex ∂y , xex ∂y , . . . , xρ1 ex ∂y , eα2 x ∂y , xeα2 x ∂y , . . . , xρ2 eα2 x ∂y , ∂x with [∂x , xk ex ∂y ] = kxk−1 ex ∂y + xk ex ∂y , [∂x , xk eα2 x ∂y ] = kxk−1 eα2 x ∂y + α2 xk eα2 x ∂y . 3.8 In general, the answer is not unique. For example, the most general valid choice for group g17 (l = 1, ρ1 = 1, α1 = 0) is U1 = (β1 γ3 − γ1 β3 )∂y , U2 = β1 ∂x + β2 ∂y + β3 x∂y , U3 = γ1 ∂x + γ2 ∂y + γ3 x∂y where β1 , . . . , γ3 are

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arbitrary constants. Choosing β1 = γ3 = 1 and 0 for the remaining coefficients leads to special case given below. g5 : U1 = ∂y , U2 = ∂x , U3 = y∂y . g7 : U1 = ∂x , U2 = ∂y , U3 = x∂x + cy∂y . g8 : U1 = ∂y , U2 = y∂y , U3 = y 2 ∂y . g10 : U1 = ∂x + ∂y , U2 = x∂x + y∂y , U3 = x2 ∂x + y 2 ∂y . g13 : U1 = ∂x , U2 = x∂x + 12 y∂y , U3 = x2 ∂x + xy∂y . g15 (r = 3): U1 = φ1 ∂y , U2 = φ2 ∂y , U3 = φ3 ∂y . g16 (r = 2): U1 = φ1 ∂y , U2 = φ2 ∂y , U3 = y∂y . g17 (l = 1, ρ1 = 1, α1 = 0): U1 = ∂y , U2 = ∂x , U3 = x∂y . g17 (l = 1, ρ1 = 1, α1 = 1): U1 = ex ∂y , U2 = xex ∂y , U3 = −∂x . g17 (l = 2, ρ1 = ρ2 = 0, α1 = 0, α2 = 1): U1 = ex ∂y , U2 = ∂y , U3 = −∂x . g17 (l = 2, ρ1 = ρ2 = 0, α1 = 1, α2 = c): U1 = ex ∂y , U2 = ecx ∂y , U3 = −∂x . g20 (r = 1): U1 = ∂x − ∂y , U2 = ∂x + ∂y , U3 = x∂x + (x + y)∂y . g24 : U1 = ∂x , U2 = ∂y , U3 = x∂x + y∂y . 3.9 The same remark applies as in the preceding exercise. g6 : U1 = −x∂x , U2 = ∂x , U3 = −y∂y , U4 = ∂y . g9 : U1 = ∂y , U2 = y∂y , U3 = y 2 ∂y , U4 = ∂x . g14 : U1 = ∂x , U2 = x∂x , U3 = x2 ∂x + xy∂y , U4 = y∂y . g15 (r = 2): Uk = φk ∂y for k = 1, . . . , 4. g16 (r = 1): U1 = ∂y , U2 = x∂y , U3 = φ(x)∂y , U4 = y∂y . g17 (l = 1, ρ1 = 2, α1 = 0): U1 = −2x∂y , U2 = 2∂y , U3 = x2 ∂y , U4 = ∂x . g17 (l = 1, ρ1 = 2, α1 = 1): U1 = ex ∂y , U2 = xex ∂y , U3 = 21 x2 ex ∂y , U4 = −∂x . g17 (l = 2, ρ1 = 0, ρ2 = 1, α1 = 0, α2 = 1): U1 = ∂x , U2 = ex ∂y , U3 = xex ∂y , U4 = ∂y . g17 (l = 2, ρ1 = 1, ρ2 = 0, α1 = 0, α2 = 1): U1 = ∂x , U2 = x∂y , U3 = ∂y , U4 = ex ∂y . g17 (l = 2, ρ1 = 1, ρ2 = 0, α1 = 1, α2 = α): U1 = ∂x , U2 = ex ∂y , U3 = xex ∂y , U4 = eαx ∂y . g17 (l = 3, ρ1 = ρ2 = ρ3 = 0, α1 = 0, α2 = 1, α3 = c): U1 = ∂x , U2 = ex ∂y , U3 = ecx ∂y , U4 = ∂y . g17 (l = 3, ρ1 = ρ2 = ρ3 = 0, α1 = 1, α2 = α, α3 = c): U1 = ∂x , U2 = ∂y , U3 = eαx ∂y , U4 = ecx ∂y . g18 (l = 1, ρ1 = 1, α1 = 0): U1 = −∂y , U2 = x∂y , U3 = ∂x , U4 = y∂y . g18 (l = 2, ρ1 = ρ2 = 0): U1 = ∂x , U2 = ex ∂y , U3 = −y∂y , U4 = ∂y . g19 (r = 2): U1 = ∂y , U2 = ∂x , U3 = x∂y , U4 = x∂x + cy∂y . g20 (r = 2): U1 = −2∂y , U2 = 2x∂y , U3 = ∂x , U4 = x∂x + (x2 + 2y)∂y . 3.10 Let the expanded form of the characteristic equation be ω(ω 3 + Aω 2 + Bω + C) = 0. For the algebra l4,4 the form of the roots entails the invariant relations B=

5 2 16 A ,

C=

1 3 32 A

366 and for l4,5 (c) B=

c2 + c − 1 2 A , 4c2

C=

c−1 3 A . 8c2

3.11 For gl2 the answer is U1 = −x∂y , U2 = 21 (y∂y − x∂x ), U3 = y∂x and U4 = x∂x + y∂y . For sl2 the generators U1 , U2 and U3 are the same as for gl2 .

4. Equivalence and Invariants of Differential Equations 4.1 Let p0 y (n) + p1 y (n−1) + . . . + pn−1 y 0 + pn y = 0 be a linear homogeneous ode with polynomial coefficients, i.e., pk ∈ Q[x]. Applying (B.16), the general transformation leading to the desired form is x = F (u), y = G(u)v with Z  1 q1 0  −1/(n+1) G = F 0n/2 q0 exp − F du . n+1 q0 F (u) is an undetermined function of u, qk (u) = pk (F ). According to Hirsch [72], p1 = −p00 is a necessary condition such that there is a fundamental system of polynomial integrals which is by no means sufficient. An example is x2 y 00 − 3xy 0 + 5y = 0 with the fundamental system y1 = x2 and y2 = x2 log x. 1 yields Choosing F = u or F = u u7/3 v 00 − 73 u4/3 v 0 −

11 1/3 v 9 u

= 0 or

u−1/3 v 00 + 13 u−4/3 v 0 −

35 −7/3 v 9 u

=0

respectively. Although q1 = −q00 , neither of these equations has a polynomial fundamental system. 4.2 A general point transformation u = σ(x, y) and v = ρ(x, y) leads from ρ v 0 = 0 to y 0 + ρxy = 0. This is the given equation y 0 + r(x, y) = 0 if there holds ρx − r(x, y)ρy = 0; i.e., ρ is the first integral of the given equation. The other transformation function σ(x, y) may be chosen arbitrarily. This result makes the degree of arbitrariness for the equation given in the proof of Theorem 4.10 completely explicit. 4.3 If F (u) = u, the transformation (4.37) simplifies to x = u, y = a12



 0 a0 a0 a00 a0 a a0 a02 v + 12 a22 − 21 a1 , y 0 = av2 − 22 v + 22 − 2a1 + 1 22 − 23 . 2 a2 2a2 2a2 a2

Substitution into (4.33) and some simplification yields a0 v 0 + v 2 + 12 z 0 − 14 z 2 + a0 a2 where z = a22 − a1 and ak ≡ ak (u). 4.4 By Theorem 4.13 the general form of the desired transformation is x = F (u), y = G(u)v + H(u). Upon substitution into Appell’s normal form the constraints F 0 G2 = 1, G0 = 0 and H = 0 are obtained. They yield the two-parameter group x = ea u + b, y = e−a/2 v with Lie algebra {∂x , ∂x − 12 ∂y }.

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4.5 Applying formulas (B.18) to Lie’s equation (4.56) yields after some rearrangement ¯ v)v 03 + B(u, ¯ v)v 02 + C(u, ¯ v)v 0 + D(u, ¯ v 00 + A(u, v) = 0; ¯ v) = −D(x, y)|x=v,y=u , B(u, ¯ v) = −C(x, y)|x=v,y=u , i.e., there holds A(u, ¯ ¯ C(u, v) = −B(x, y)|x=v,y=u and D(u, v) = −A(x, y)|x=v,y=u . 4.6 Applying the transformation x = αu, y = eβu v to y 000 + cy 0 + y = 0 yields v 000 + 3βv 00 + (3β 2 + cα2 )v 0 + (α3 + α2 βc + β 3 )v = 0. Retaining the structure of the original equations requires β = 0 and α3 = 1. From the factorization α3 − 1 = (α − 1)(α2 − α − 1) the asserted equivalence is obvious. 4.7 If y 000 − (a + 1)y 00 + ay 0 = 0 is transformed by x = αu, y = eβu v, the equation v 000 +(3β−aα−α)v 00 +(3β 2 +aα2 −2aαβ−2αβ)v 0 +(β 2 +aα2 −aαβ−αβ)βv = 0 is obtained. In order to retain the structure of the original equation, the coefficient of v must vanish. This yields β(α − β)(aα − β) = 0; i.e., β = 0, β = α or β = aα. Substituting these values into the coefficients of v 00 and v 0 and imposing the conditions for the structural invariance leads to the given five alternatives. 4.8 By Theorem 5.15, case ii), the only nonvanishing coefficients of equation (5.41) are B3 = 2p(f ) and B5 = 2q(u). Consequently, it belongs to symmetry class S63 . The invariant J1 = 2p(f ) in equations (6.72) or (6.73) shows that determining S or R comes down to solving a linear second order equation. If the goal is to solve any given third order equation in S63 this is an important simplification. In the particular application at issue, however, it does not lead toward the desired answer. 4.9 Let the given equation be in rational normal form y 00 + q(x)y = 0 with q(x) ∈ Q(x). In order to decide the desired equivalence, a rational solution of the equation  f 00 0 1  f 00 2 1 (A.5) − f 02 f 2 − (a − 1)f 02 = 2q(x) − 2 2 f0 f0 for f (x) has to be found. In a partial fraction expansion of f let the leading term for x → ∞ be aN xN . The first two terms at the left hand side generate 2 a term 1 − 2N from it if N > 1, whereas the remaining two terms generate x the leading powers 4N − 2 and 2N − 2 for N ≥ 1. If n is the leading power in the partial fraction expansion of q(x), the leading power 4N − 2 must be 1 matched by it which leads to the bound N = n + 4 for the behavior at infinity. For any pole of order M at x = x0 , a similar consideration yields the term 1 − M 2 for M > 1 and two more terms of highest power 4M +2 and 2M for (x − x0 )2

368 M ≥ 1. The leading power 4M + 2 must be matched by a singularity of order m such that M = m 4− 2 . Consequently, all singularities of a rational solution are determined by the function q(x) at the right hand side and its orders are properly bounded. An ansatz with undetermined coefficients within these bounds leads to a linear system any solution of which determines a rational solution for f . 4.10 With the same notation as above, a rational solution has to be found for   f 00 0 1  f 00 2 1  f 0 2 − −2 ν 2 − +2f 02 = 2q(x). 0 0 2 4 f f f

(A.6)

By a similar argumentation, the bounds for the polynomial part and any finite m pole are now N = n 2 + 1 and M = 2 − 1 respectively leading again to a linear system for the undetermined coefficients.

5. Symmetries of Differential Equations 5.1 By Definition 5.3 the left hand side of the relation at issue may be written as U (n+1) D(φ) = ξ

∂D(φ) ∂D(φ) ∂D(φ) +η + ζ (1) + ... ∂x ∂y ∂y 0 ∂D(φ) ∂D(φ) +ζ (n) + ζ (n+1) (n+1) . ∂y (n) ∂y

Expanding the right hand side yields D(U (n) ) − D(ξ)D(φ)   ∂φ ∂φ ∂φ ∂φ +η + ζ (1) 0 + . . . + ζ (n) (n) −D(ξ)D(φ) =D ξ ∂x ∂y ∂y ∂y ∂φ ∂φ ∂φ (1) ∂φ = D(ξ) + D(η) + D(ζ ) 0 + . . . + D(ζ (n) ) (n) ∂x ∂y ∂y ∂y       ∂φ ∂φ ∂φ +ξD +ηD +ζ (1) D +... 0 ∂x ∂y ∂y   ∂φ +ζ (n) D −D(ξ)D(φ). ∂y (n) Applying the relations D(η) = ζ (1) + y 0 D(ξ),

D(ζ (k) ) = ζ (k+1) + y (k+1) D(ξ)

(A.7)

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for k = 1, . . . , n that follow from (5.10) it may be rewritten as D(ξ)

 ∂φ  (2)  ∂φ ∂φ  (1) + ... + ζ + y 0 D(ξ) + ζ + y 00 D(ξ) ∂x ∂y ∂y 0   ∂φ + ζ (n+1) + y (n+1) D(ξ) ∂y (n)       ∂φ ∂φ ∂φ +ηD +ζ (1) D +ξD +... ∂x ∂y ∂y 0   ∂φ +ζ (n) D −D(ξ)D(φ). ∂y (n)

The first term and the second term in the square brackets combined cancel the last term with the result     i h  ∂φ ∂φ ∂φ ∂φ +ηD +ζ (1) +D +... ξD 0 ∂x ∂y ∂y ∂y  h i ∂φ ∂φ +ζ (n+1) . (n) + D (n+1) ∂y ∂y The terms in the square brackets may be simplified using the obvious relations ∂x D = D∂x , ∂y D = D∂y , ∂y(k) D = ∂y(k−1) + D∂y(k) for k = 1, . . . n such that they yield the right hand side of (A.7). 5.2 Applying Theorem 5.2 to y 00 + r(x)y = 0 leads to the determining system ξyy = 0, ηyy − 2ξxy = 0, ηxy − 21 ξxx + 23 ryξy = 0, ηxx − ry(ηy − 2ξx ) + rη + r0 yξ = 0. Its general solution contains eight constants and yields the generators U1 = y1 ∂y , U2 = y2 ∂y , U3 = y∂y , U4 = y1 y∂x + y10 y 2 ∂y , U5 = y2 y∂x + y20 y 2 ∂y , U6 = y12 ∂x + y1 y10 y∂y , U7 = y22 ∂x + y2 y20 ∂y , U8 = 2y1 y2 ∂x + (y10 y2 + y1 y20 )y∂y . 5.3 If a symmetry generator is searched for in the form U = η(x)∂y , (5.10) ∂ (k) η yields ζ (k) = . Consequently, the determining system for η(x) is ∂xk p0 η + p1 ηx + p2 ηxx + . . . +

∂ (n) η = 0; ∂xn

i.e., it is identical to the given equation for y. Now if a symmetry generator is searched for in the form U = η(y)∂y , from equations (5.10) it follows ζ (1) =

370 y 0 ηy and ζ (k) = D(ζ (k−1) ) with D = y 0 ∂y + y 00 ∂y0 + . . ., therefore the general structure of ζ (k) is ∂kη ζ (k) = y 0 k + y (k) ηy + . . . ∂y ∂iη with 1 < i < ∂y i = 0, pn η − pn yηy = 0 follows

where the omitted terms are proportional to derivatives

n − 1. From this the determining system ηyy with the solution η = y. 5.4 The most general form that is linear and homogeneous in u, v and w is wx + u + P v + Qw = 0,

wy + Ru − v + Sw = 0.

The requirement that no new integrability conditions are introduced is assured under the condition Sx − Qy − 31 Cy − 13 Bx = 0. 5.5 The subsequent listing allows one to associate Lie’s enumeration of groups 1), . . . , 13) with the notation in this book: 1) ≡ g10 , 2) ≡ g13 , 3) ≡ g8 , 4) ≡ g7 , 5) ≡ g17 with l = 2, ρ1 = ρ2 = 0, α1 = 1, α2 = α 6= 0, 1, 6) ≡ g24 , 7) ≡ g16 , 8) ≡ g20 , 9) ≡ g17 with l = 1, ρ1 = 1, α1 = 1, 10) ≡ g5 , 11) ≡ g17 with l = 2, ρ1 = ρ2 = 0, α1 = 0, α2 = 1, 12) ≡ g17 with l = 1, ρ1 = 1, α1 = 0, 13) ≡ g15 . For three groups a change of variables has to be applied in order to obtain the generators in Lie’s form: 9) is obtained from g17 by y¯ the change x = −¯ x, y = y¯ex¯ , 11) by the change x = − log x ¯, y = x ¯ and 5) 1 1/(α−1) by x = α − 1 log x ¯ and y = y¯x ¯ . Finally, the barred variables have to be replaced by the original ones. In Theorem 5.8 there occur only groups 1), 2), 4) and 8) because there are no second order ode’s with a symmetry group different from these. This is related to the definition of a symmetry group as the maximal group of transformations leaving the equation invariant; see also the remarks by Lie [113] on page 538. 5.6 In order to avoid an unspecified dependence on y in the transformed 00 function r, σy = 0 is required; i.e., σ(x, y) ≡ f (x). The quotient v 0 is v reproduced if the two derivatives are proportional to y 00 and y 0 respectively. This is assured if ρx = 0 and ρyy = 0, i.e., ρ = a1 y + a2 . It yields v 00 1 y 00 f 00 a1 000 3a1 f 00 00 3a1 (f 002 − f 000 f 0 ) 0 000 = − and v = y − y + y. v0 f 0 y0 f 02 f 03 f 04 f 05 If the second term of the expression for v 000 is rewritten as −

3a1 f 00 y 00 0 y its f 04 y 0

correct structure is obvious. 5.7 In order to avoid an unspecified dependence on y in the transformed function r, σy = 0 is required, i.e., σ = h(x). The independence of v 00 on y 0 in the first place requires ρyy = 0 due to the quadratic term in equation (5.5), i.e., it has the form ρ(x, y) = f (x) + g(x)y. The coefficients of the linear term

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in y 0 and the independence on y vanish if f and g satisfy 2

h00 g0 − 0 = 0, g h

h00 g 00 − 0 =0 0 g h

with the general solution g = C x 1+ C , h = C3 g + C4 , and, consequently, 1 2 σ=

(C1 x + C2 )C4 + C3 , C1 x + C2

ρ=

y . C1 x + C2

The function f (x) remains undetermined. Introducing new constants by C1 = −

a2 a4 1 − a1 a2 a4 1 a3 , C2 = , C3 = , C4 = − a3 a3 a4 a4

yields the required answer. 5.8 Both conditions are subsumed under the constraint Ψ03 = 0 with Ψ3 defined in Theorem 4.9 as a simple calculation shows. They may also be obtained from the Janet basis for the symmetry generators as it has been shown by Schwarz [166]. 5.9 Equation (5.46) is completely reducible into first order factors 3 + 3, D − 1 − 1 − 2 + 3, Lclm(D − 23 x − 2 x 2 x+2 x−2 x 1 2 1 + 3 , D + 3 − 3 + 3 )y = 0 D+ 2 − x+2 − x− 2 x 2 x+2 x where D ≡ d . From this representation the answer is obvious. dx

6. Transformation to Canonical Form 6.1 Denoting the symmetry algebra of symmetry class S12 by L21 , symmetry 2 class S2,1 by L22,1 , etc., the diagram  L21  @   R 2 @ 2 L2,1 L2,2   @ 
   R 2 @ L23,4 L3,3
L23,2 L23,1
    PP 
 PP 
 P  q P ? ) L28  describes the relationship between the various Lie algebras. An arrow from Li to Lj means on the one hand that Li ⊂ Lj . On the other hand it implies

372 that the undetermined elements in the canonical form corresponding to the former symmetry algebra may be specialized such that the symmetry algebra enlarges to the latter. Accordingly, the arrows originating from L21 imply the relations ru (u, v 0 ) = 0 or uru (u, v 0 ) + r(u, v 0 ) = 0, respectively, for the undetermined function r(u, v 0 ) in the canonical form for the symmetry class S12 as given in Theorem 6.4. The arrows originating from L22,1 imply that the 2 undetermined function r(v 0 ) in the canonical form for symmetry class S2,1 2 2 has the special form required for symmetry class S3,3 or S3,4 . The remaining arrows imply that the respective undetermined elements vanish such that the canonical form v 00 = 0 for symmetry class S82 is obtained. 6.2 The desired transformation must have the form u = a1 u ¯ + a2 v¯ + a3 , v = a4 u ¯ + a5 v¯ + a6 . Substituting it into the canonical form of Example 6.7 leads to the algebraic system a1 (4a31 + 8a21 a4 + 5a1 a24 + a34 ) = 0, a2 (4a32 − 8a22 a4 + 5a2 a24 − a34 ) = 0, (a1 + a2 )a24 = 1. There are 24 alternatives. A particularly simple one is a1 = 0, a2 = a4 = 1, a5 = −1, a3 and a6 arbitrary. 2 6.3 For symmetry class S2,1 the system (6.4) may be solved for the Janet basis coefficients with the result a1 = b2 = I3 , a2 = I4 , b1 = I1 , c1 = d2 = −I5 , c2 = −I6 , d1 = −I2 . These relations have obviously the form given in Theorem 6.6. For symmetry 2 class S2,2 , system (6.9) yields a1 = −I3 − K2 , a2 = −I4 − K1 , b1 = −I1 , b2 = −I3 , c1 = I5 , c2 = I6 , d1 = I2 − K2 , d2 = I5 − K1 . They may be rewritten in the form given in Theorem 6.6 as follows. I1 = −b1 , I2 = d1 − a1 + b2 , I3 = −b2 , I4 = d2 − a2 − c1 , I5 = c1 , I6 = c2 , K1 = c1 − d2 , K2 = b2 − a1 . 6.4 Applying the algorithm Symmetries on page 199 to y 00 + r(y, y 0 ) = 0 generates the Janet basis η = ξx = ξy = 0; i.e., case b) of Theorem 6.7 with a = b = c = 0. Consequently, φ = y and G = 0, and the general transformation may be written in the form x = Cv + f (u), y = g(u) with suitably defined functions f and g. Together with y0 =

g0 , Cv 0 + f 0

y 00 =

−Cg 0 v 00 + Cg 00 v 0 + f 0 g 00 − f 00 g 0 (Cv 0 + f 0 )2

the desired transformation follows. The special values C = 1, f = 0 and g = u generate the variables exchange discussed after the proof of Theorem 6.7.

Solutions to Selected Problems

373

2 ξ = 0, ξ = 0, η = 0 and 6.5 The vector fields determined by ξx − x y x 2 ηy − y η = 0 generate a two-parameter subgroup of the full symmetry group because it reduces the Janet basis (5.32) to zero. Applying the scheme de1 σ = 0, σ = 0, σ + 2 σ = 0 is scribed in Theorem 6.8, the system σxx + x xy yy x y y  1 1 1 obtained with the fundamental system 1, x , y . The transformation x = u and y = v1 yields v 00 v 0 −1 = 0; i.e., a canonical form v 00 +r(v 0 ) = 0 correspond2 ing to symmetry class S2,1 . In order to obtain the special value r(v 0 ) = v 04 , the genuine symmetry class has to be known. η+ξ 6.6 The vector fields determined by ξx − x + y = 0, ξy = 0, ηx = 0 and η+ξ ηy − x + y η = 0 generate a two-parameter subgroup of the full symmetry group because it reduces the Janet basis given in Example 5.14 to zero. Evaluating the integrals given in Theorem 6.9 yields for the transformation functions σ = x + y and ρ = y. Substituting x = u − v and y = v into the given equation yields the canonical form uv 00 − 2v 03 − 3v 02 − v 0 = 0. The canonical form given in Example 6.11 cannot be obtained by specialization from it. 6.7 Three generators of a sub-algebra of the full symmetry algebra are U1 = 12 x∂x , U2 = x2 (y − 1)2 ∂y and U3 = y(y − 1)∂y with commutators [U1 , U2 ] = U2 , [U1 , U3 ] = 0 and [U2 , U3 ] = −U2 . A g26 subgroup is generated by U1 and U3 . From its Janet basis

ξx +

1 2y − 1 ξ = 0, ξy = 0, ηx = 0, ηy − η=0 x y(y − 1)

1 σ = 0, σ = 0, σ + 2y − 1 σ = 0 with fundaTheorem 6.8 yields σxx + x x xy yy y(y − 1) y  y − 1 mental system 1, log x, log y . It leads to the transformation functions y−1 1 u = log x and v = log y with the inverse x = eu , y = from which 1 − ev 00 02 0 the canonical form v − v − 3v − 1 = 0 is obtained. Choosing the two-parameter group of type g25 generated by U1 and U2 with 1 ξ = 0, ξ = 0, η − 2 η = 0, η − 2 η = 0, by Theorem 6.8 Janet basis ξx − x y x y x y−1 1 1 there follows u = 2 , v = 2 . From its inverse x = √1 and y = u v +1 u x x (y − 1) the canonical form uv 00 + 21 v 0 + 12 = 0 follows, i.e., an inhomogeneous linear ode. The three generators U1 , U2 and U3 generate a three-parameter group with Lie algebra l3,4 . This symmetry type is not included in the classification given in Theorem 5.8 because by definition a symmetry class is defined by the maximal group leaving it invariant.

374

7. Solution Algorithms 7.1 For the ode in Example 6.2 the equation dy + (y 2 − 2x2 y + x4 − 2x − 1)dx =0 y 2 − 2x2 y + x4 − 1 is obtained. Upon integration it yields the same answer as given there. In this case, the integrations for obtaining the canonical form are simpler. 1 7.2 In Example 6.2 the transformation functions x ≡ φ(u, v) = u − v + 2v 1 have been shown to generate the canonical and y ≡ ψ(u, v) = u − v − 2v 1 + x + y , v = 1 into the form v 0 = 13 . Substituting its inverse u = x − y x−y 2 canonical form solution v = 23 u + C yields the solution in actual variables as given in Example 6.2. 7.3 According to Lemma 3.4, canonical coordinates are u = y 2 −x2 and v = y+x 1 1 u+1 0 2 log y − x . They lead to the canonical form v + 2u u − 1 = 0. Integrating and resubstituting x and y yield the solution given in Example 6.2. 7.4 Let x = g(y 0 ) be a first order ode and consider y 0 as aR parameter, say p, such p = y 0 = h(x) with h = g −1 . It follows y = h(x)dx + C or R that R 0 y = pg (p)dp + C. Partial integration yields y = pg(p) − g(p)dp + C. Substituting u and v for x and y respectively, −R(p) + C1 for g, and C2 for C leads to (7.16). ¯ ≡ w(t), ¯ ¯0 = w 7.5 Substituting u = ew¯ , z = t with w and then w ¯ into (7.38) 3 0 2 3 2 yields w ¯ + t(t + 4)w ¯ + 6w ¯ + tw ¯ = 0. This is an Abel equation. Its rational normal form w0 + t(t2 + 4)w3 + 2 3t w = 0 is obtained by substituting (see t +4 the proof of Theorem 4.14) w ¯ = w − 22 and leaving t unchanged. This t(t + 4) Bernoulli equation has the general solution − 1  w = C(t2 + 4)3 + 14 (t2 + 4)2 t2 2 . Resubstitution of the original variables and redefining the integration constants finally yields the solution given in Example 7.19. R p(p + 1)2 p+1 2p 7.6 Now r(p) = , R(p) = log + and R(p)dp = p p−1 p+1 (p − 1) log (p + 1) − p log p + 2p. Substitution into (7.16) yields u + log

p+1 2p 2p + = C¯1 , v + log (p + 1) − = C¯2 p p+1 p+1

and finally x=

C1 2p C2 p 2p exp , y= exp − . p+1 p+1 p+1 p+1 (2,2)

7.7 For generic r(y), a type J1,2 Janet basis with a = b = c = 0 is obtained for the determining system. By Theorem 6.7, case b), the transformation to canonical form is achieved by x = v, y = u with the result

Solutions to Selected Problems

375

v 00 − v 03 r(u) = 0. Two integrations and resubstitution of the actual variables leads to the integral representation Z dy q = x + C2 R C1 − 2 r(y)dy of the general solution. (2,2) 7.8 For generic r(x), a type J1,1 Janet basis with a = 0, b = − y1 is obtained for the determining system. By Theorem 6.7, case a), the transformation to canonical form is achieved by x = u, y = ev with the result v 00 + r(u)v 02 + v 0 = 0. This Riccati equation for v 0 has the general solution v0 = R

e−u . r(u)e−u du + C1

Integration and resubstitution leads to the general solution in actual variables Z e−x dx y = C2 exp R . r(x)e−x dx + C1 7.9 The first integrals following from the equivalent pde may be found in Lie [113], page 540. In the notation applied in this book they yield the relations u2 +

0 v 2 − 2uvv 0 ¯1 , u − vv ¯ = C a a = C2 . v 02 − 2 v 02 − 2 v v

Elimination of v 0 leads to u4 − 4C¯1 u3 + (4C¯12 − 2C¯2 )u2 + 4C¯1 C¯2 u + C¯22 v4 − a = 0. C¯12 + C¯2 2 This representation is equivalent to (7.25) if C12 = a¯ + C¯2 , C2 = −C¯1 . C1 This result shows that the solutions obtained from the first integrals of the equivalent pde may be unnecessarily complicated. Determine the solution of (7.24) from the equivalent pde and compare the result with (7.25). 1 = 0. The sym7.10 Write the equation to be solved as y 0 + 1√ y 2 + √ ax x a x y metry generator x∂x + 12 y∂y yields the canonical variables u = √ , v = 2 log y. x 2 1 4 u v v/2 0 Substituting the inverse x = 2 e , y = e yields v = u 2 + 1 . An u 2u + au + 2 4ai 4u + a integration leads to v = 2 log u + p arctan p + C. Applying a2 − 16 i a2 − 16 ix the relation arctan x = − 2i log 11 + − ix , it assumes the form p 2a a2 − 16 + a + 4u v = 2 log u + p log p + C. a2 − 16 a2 − 16 − a − 4u

376 Resubstitution of the original variables and some simplifications lead to √ a x (c − 12 )xc − (c + 12 )C y= 2 xc + C p 2 where c = a 2a− 16 . 7.11 The pde for (7.71) is Af = 0 with A = ∂u + v 0 ∂v + v 00 ∂v0 + ω∂v00 and √ 00 ω = − vv (3v 0 + a v 00 v). The linear relation between A and the four vector fields generating the symmetry group yields the first integrals √ √ 2 0 00 4vv 0 + av v 00 v 2vv 0 2 2v − (4vv + av v v)u , Φ2 = 2u− , Φ3 = u + Φ1 = u− D D D √ with D = 2vv 00 + 2v 02 + av 0 v 00 v. Solving the relations Φi = Ci , i = 1, 2, 3, 0 for vv yields the integrand in (7.71).

Appendix B Collection of Useful Formulas

For ease of reference in this appendix those formulas are put together that occur frequently in the main text. Moreover, they might be of general interest for any person working in this field. Transformations of the Plane. Let a coordinate transformation from x, y to u, v and its inverse be defined by x = φ(u, v),

y = ψ(u, v)

u = σ(x, y), v = ρ(x, y).

(B.1)

They entail the following relations for the first order partial derivatives.    −1   1 σx σy φu φv ψv −φv (B.2) = = ρx ρy ψu ψv φu ψv − φv ψu −ψu φu and 

φu φv ψu ψ v



 =

σx σy ρx ρy

−1

1 = σ x ρy − σ y ρx



ρy −σy −ρx σx

 .

(B.3)

It is assumed that ∆ ≡ σx ρy − σy ρx 6= 0 and φu ψv − φv ψu 6= 0. The first and second order derivatives of φ and ψ are φu =

ρy −σy −ρx σx , φv = , ψu = , ψv = , ∆ ∆ ∆ ∆

(B.4)

φuu = 13 [ρy (ρxy ∆ − ρy ∆x ) − ρx (ρyy ∆ − ρy ∆y )], ∆ φuv = 13 [−σy (ρxy ∆ − ρy ∆x ) + σx (ρyy ∆ − ρy ∆y )], ∆ φvv = 13 [σy (σxy ∆ − σy ∆x ) − σx (σyy ∆ − σy ∆y )], ∆ (B.5) ψuu = 13 [−ρy (ρxx ∆ − ρx ∆x ) + ρx (ρxy ∆ − ρx ∆y )], ∆ ψuv = 13 [σy (ρxx ∆ − ρx ∆x ) − σx (ρxy ∆ − ρx ∆y )], ∆ ψvv = 13 [−σy (σxx ∆ − σx ∆x ) + σx (σxy ∆ − σx ∆y )]. ∆ If y ≡ y(x) and v ≡ v(u), the substitutions (B.1) generate the following transformation of the derivatives y 0 , y 00 and y 000 . y0 =

ψ u + ψv v 0 , φu + φv v 0

(B.6)

377

378 y 00 =

 1 00 03 0 3 (φu ψv − φv ψu )v + (φv ψvv − φvv ψv )v (φu + φv v )   + φu ψvv − φvv ψu + 2(φv ψuv − φuv ψv ) v 02

  + φv ψuu − φuu ψv + 2(φu ψuv − φuv ψu ) v 0 + φu ψuu − φuu ψu . (B.7) Higher order derivatives are obtained from (5.6). In general the structure of the kth derivative is h m φ (φ ψ − φv ψu )v 00 v (k−1) 1 (φu ψv − φv ψu )v (k) − k v u v y (k) = 0 k+1 φu + φv v 0 (φu + φv v ) i P (v 0 )v (k−1) + Q(v 0 , . . . , v (k−2) . + 0 k−2 (φu + φv v ) ) (B.8) P and Q are polynomial in their arguments and furthermore depend on u and v; mk is a natural number. If φv = 0, this expression simplifies to h i F (u, v)v (k−1) + G(v 0 , . . . , v (k−2) ) y (k) = 1k ψv y (k) + kv 0 v (k−1) + . φu φk−1 u

(B.9)

The expressions (B.8) and (B.9) are due to Neumer [137]. Various special cases that occur frequently in this book lead to simplifications of these expressions. If x = F (u), y = G(v), F 0 =

dF dG and G0 = du dv

(B.10)

the derivatives are 0 y 0 = G0 v 0 , y 00 = 103 (F 0 G0 v 00 + F 0 G00 v 02 − F 00 G0 v 0 ), F F 1 000 02 0 000 y = 05 (F G v + 3F 02 G00 v 0 v 00 + F 02 G000 v 03 − 3F 00 F 0 G0 v 00 F −3F 00 F 0 G00 v 02 + 3F 002 G0 v 0 + F 000 F 0 G0 v 0 ).

(B.11)

If x = F (u), y = G(u)v + H(u), F 0 =

dG dH dF , G0 = and H 0 = (B.12) du du du

the derivatives are   0 y 0 = G0 v 0 + G v + H , G F h   0 0 00 y 00 = G02 v 00 + 2 G F − F F 02 v 0 G F  00  i G0 F 00 F 02 v + (H 00 F 0 − H 0 F 00 ) F 02 . 0 + G F − G G G (B.13)

Collection of Useful Formulas If x = F (u), y = G(u)v, F 0 =

dF dG and G0 = du du

379

(B.14)

the derivatives are  0  y 0 = G0 v 0 + G Gv , F h     i 0 F 00 v 0 + G00 − F 00 G0 v , y 00 = G02 v 00 + 2 G − 0 0 G G F F G F  0 n 00  F 00 y 000 = G03 v 000 + 3 G G − F0 v F i h  00 2 00 F 00 G0 − F 000 v 0 − 6 + 3 F0 +3 G 0 0 G F F G F h  00 2 0 i o G + G000 − F 000 G0 − 3 F 00 G00 v . + 3 F0 0 0 G G F F G F G (B.15) For some applications the explicit form of the coefficients of the three leading derivatives are required. They may be seen from n h 0 i n − 1 F 00 v (n−1) y (n) = G0n v (n) + n G − 0 2 F G F h 00 n(n − 1) G F 00 G0 + − (n − 1) 2 G F0 G  2 i o 2 F 000 + (n − 2)(n + 1) F 00 (n−2) −n − v +... 0 0 3 F 4 F (B.16) Upon further specialization G = F 0 , it simplifies to n n(n − 3) F 00 (n−1) 1 v y (n) = 0n−1 v (n) − 2 F0 F h i o n(n − 1) n − 5 F 000 n2 − 5n + 2 ( F 00 )2 v (n−2) + . . . − − 0 0 2 3 F 4 F (B.17) The independent variable x and the dependent variable y ≡ y(x) are exchanged by the transformation x = v, y = u with v ≡ v(u). The derivatives up to order four are transformed by 00 000 002 y 0 = 10 , y 00 = − v03 , y 000 = − v 04 + 3v05 , v v v v (4) 000 00 003 y (4) = − v 05 + 10v 06 v − 15v06 . v v v

(B.18)

Symmetric Powers of Second Order Equations. The listing below gives the k-th symmetric powers of y 00 + ry = 0 from k = 2 to k = 6. k=2:

y 000 + 4ry 0 + 2r0 y = 0,

380 y (4) + 10ry 00 + 10r0 y 0 + (3r00 + 9r2 )y = 0,

k=3: k=4:

y (5) + 20ry 000 + 30r0 y 00 + (18r00 + 64r2 )y 0 + (4r000 + 64r0 r)y = 0,

k=5:

y (6) + 35ry (4) + 70r0 y 000 + (63r00 + 259r2 )y 00 + (28r000 + 518r0 r)y 0 2

+(5r(4) + 155r00 r + 130r0 + 225r3 )y = 0, y (7) + 56ry (5) + 140r0 y (4) + (168r00 + 784r2 )y 000 + (112r000 + 2352r0 r)y 00

k=6:

2

+(40r(4) + 1408r00 r + 1180r0 + 2304r3 )y 0 +(6r(5) + 312r000 r + 708r00 r0 + 3456r0 r2 )y = 0, Janet Basis Types. The Janet basis types that are required in earlier parts of this book are listed, beginning with the Janet bases of type J (1,2) of order 1, 2 and 3. Their coherence conditions may be found on page 59. The dependent variable is z, the independent variables are x and y, the grlex term order with y > x is always applied. (1,2)

J1 (1,2)

J2,1

:

(1,2)

(1,2)

(1,2)

J3,2

:

:

zx + az = 0, zy + bz = 0.

zy + a1 zx + a2 z = 0, zxx + b1 zx + b2 z = 0.

J2,2 J3,1

:

:

zx + a1 z, zyy + b1 zy + b2 z = 0.

zy + a1 zx + a2 z = 0, zxxx + b1 zxx + b2 zx + b3 z = 0.

zxx + a1 zy + a2 zx + a3 z = 0, zxy + b1 zy + b2 zx + b3 z = 0, zyy + c1 zy + c2 zx + c3 z = 0. (1,2)

J3,3

:

zx + a1 z = 0, zyyy + b1 zyy + b2 zy + b3 z = 0.

The Janet bases of type J (2,2) of order 1, 2 and 3 are given next. Their coherence conditions may be found on pages 64 to 67. The dependent variables are z1 and z2 . The term order is grlex with z2 > z1 and y > x. (2,2)

J1,1 (2,2)

J1,2 (2,2)

J2,1

:

(2,2)

J2,3

:

z1 = 0, z2,x + az2 = 0, z2,y + bz2 = 0.

z2 + az1 = 0, z1,x + bz1 = 0, z1,y + cz1 = 0.

z1 = 0, z2,y + a1 z2,x + a2 z2 = 0, z2,xx + b1 z2,x + b2 z2 = 0.

(2,2)

J2,2

:

:

:

z1 = 0, z2,x + a1 z2 = 0, z2,yy + b1 z2,y + b2 z2 = 0. z1,x + a1 z2 + a2 z1 = 0, z1,y + b1 z2 + b2 z1 = 0, z2,x + c1 z2 + c2 z1 = 0, z2,y + d1 z2 + d2 z1 = 0.

Collection of Useful Formulas (2,2)

J2,4

:

z2 + a1 z1 = 0, z1,y + b1 z1,x + b2 z1 = 0, z1,xx + c1 z1,x + c2 z1 = 0.

(2,2)

J2,5 (2,2)

J3,1

:

:

z2 + a1 z1 = 0, z1,x + b1 z1 = 0, z1,yy + c1 z1,y + c2 z1 = 0.

z1 = 0, z2,y + a1 z2,x + a2 z2 = 0, z2,xxx + b1 z2,xx + b2 z2,x + b3 z2 = 0. (2,2)

J3,2 (2,2)

J3,3

(2,2)

J3,4

381

:

:

z1 = 0, z2,xx + a1 z2,y + a2 z2,x + a3 z2 = 0, : z2,xy + b1 z2,y + b2 z2,x + b3 z2 = 0, z2,yy + c1 z2,y + c2 z2,x + c3 z2 = 0.

z1 = 0, z2,x + a1 z2 = 0, z2,yyy + b1 z2,yy + b2 z2,y + b3 z2 = 0.

z1,x + a1 z2 + a2 z1 = 0, z1,y + b1 z2 + b2 z1 = 0, z2,y + c1 z2,x + c2 z2 + c3 z1 = 0, z2,xx + d1 z1,x + d2 z2 + d3 z1 = 0.

(2,2)

J3,5

:

z1,x + a1 z2 + a2 z1 = 0, z1,y + b1 z2 + b2 z1 = 0, z2,x + c1 z2 + c2 z1 = 0, z2,yy + d1 z2,y + d2 z2 + d3 z1 = 0.

(2,2)

:

z1,y + a1 z1,x + a2 z2 + a3 z1 = 0, z2,x + b1 z1,x + b2 z2 + b3 z1 = 0, z2,y + c1 z1,x + c2 z2 + c3 z1 = 0, z1,xx + d1 z1,x + d2 z2 + d3 z1 = 0.

(2,2)

:

z1,x + a1 z2 + a2 z1 = 0, z2,x + b1 z1,y + b2 z2 + b3 z1 = 0, z2,y + c1 z1,y + c2 z2 + c3 z1 = 0, z1,yy + d1 z1,y + d2 z2 + d3 z1 = 0.

J3,6

J3,7 (2,2)

:

z2 + a1 z1 = 0, z1,x + b1 z1 = 0, z1,yyy + c1 z1,yy + c2 z1,y + c3 z1 = 0.

(2,2)

:

z2 + a1 z1 = 0, z1,xx + b1 z1,y + b2 z1,x + b3 z1 = 0, z1,xy + c1 z1,y + c2 z1,x + c3 z1 = 0, z1,yy + d1 z1,y + d2 z1,x + d3 z1 = 0.

J3,8

J3,9

(2,2)

J3,10 :

z2 +a1 z1 = 0, z1,y +b1 z1,x +b2 z1 = 0, z1,xxx +c1 z1,xx +c2 z1,x +c3 z1 = 0.

A partial listing of Janet basis types of order four is given next; they are required for the symmetry class identification on page 227. (2,2) J4,2

(2,2) J4,9

z1 = 0, z2,xy + a1 z2,xx + a2 z2,y + a3 z2,x + a4 z2 = 0, : z2,yy + b1 z2,xx + b2 z2,y + b3 z2,x + b4 z2 = 0, z2,xxx + c1 z2,xx + c2 z2,y + c3 z2,x + c4 z2 = 0.

z1,y + a2 z1,x + a3 z2 + a4 z1 = 0, z2,y + b1 z2,x + b2 z1,x + b3 z2 + b4 z1 = 0, z1,xx + c1 z2,x + c2 z1,x + c3 z2 + c4 z1 = 0, : z2,xx + d1 z2,x + d2 z1,x + d3 z2 + d4 z1 = 0.

382 (2,2)

J4,10 :

z1,y + a2 z1,x + a3 z2 + a4 z1 = 0, z2,x + b2 z1,x + b3 z2 + b4 z1 = 0, z1,xx + c1 z2,y + c2 z1,x + c3 z2 + c4 z1 = 0, z2,yy + d1 z2,y + d2 z1,x + d3 z2 + d4 z1 = 0.

(2,2) J4,12

z1,x + a3 z2 + a4 z1 = 0, z2,x + b2 z1,y + b3 z2 + b4 z1 = 0, z1,yy + c1 z2,y + c2 z1,y + c3 z2 + c4 z1 = 0, : z2,yy + d1 z2,y + d2 z1,y + d3 z2 + d4 z1 = 0.

(2,2)

J4,14

(2,2) J4,17

z2,x + a1 z1,y + a2 z1,x + a3 z2 + a4 z1 = 0, z2,y + b1 z1,y + b2 z1,x + b3 z2 + b4 z1 = 0, z : 1,xx + c1 z1,y + c2 z1,x + c3 z2 + c4 z1 = 0, z1,yy + d1 z1,y + d2 z1,x + d3 z2 + d4 z1 = 0, z1,yy + e1 z1,y + e2 z1,x + e3 z2 + e4 z1 = 0.

z2 + a4 z1 = 0, z1,xy + b1 z1,xx + b2 z1,y + b3 z1,x + b4 z1 = 0, z1,yy + c1 z1,xx + c2 z1,y + c3 z1,x + c4 z1 = 0, : z1,xxx + d1 z1,xx + d2 z1,y + d3 z1,x + d4 z1 = 0.

(2,2) J4,19

z2 = 0, z1,xx + a2 z1,y + a3 z1,x + a4 z1 = 0, z1,xy + b2 z1,y + b3 z1,x + b4 z1 = 0, : z1,yyy + c1 z1,yy + c2 z1,y + c3 z1,x + c4 z1 = 0.

Finally, the following fifth order Janet bases are required for symmetry class identification on page 238.

(2,2)

J5,1

(2,2)

J5,2

(2,2)

J5,3

z1,x + a4 z2 + a5 z1 = 0, z1,y + b4 z2 + b5 z1 = 0, z2,xy + c1 z2,xx + c2 z2,y + c3 z2,y + c4 z2 + c5 z1 = 0, : z2,yy + d1 z2,xx + d2 z2,y + d3 z2,y + d4 z2 + d5 z1 = 0, z2,xxx + e1 z2,xx + e2 z2,y + e3 z2,y + e4 z2 + e5 z1 = 0. z2,x + a2 z1,y + a3 z1,x + a4 z2 + a5 z1 = 0, z2,y + b2 z1,y + b3 z1,x + b4 z2 + b5 z1 = 0, : z1,xx + c1 z1,yy + c2 z1,y + c3 z1,x + c4 z2 + c5 z1 = 0, z1,xy + d1 z1,yy + d2 z1,y + d3 z1,x + d4 z2 + d5 z1 = 0, z1,yyy + e1 z1,yy + e2 z1,y + e3 z1,x + e4 z2 + e5 z1 = 0. z2,x + a2 z1,y + a3 z1,x + a4 z2 + a5 z1 = 0, z2,y + b2 z1,y + b3 z1,x + b4 z2 + b5 z1 = 0, : z1,xy + c1 z1,xx + c2 z1,y + c3 z1,x + c4 z2 + c5 z1 = 0, z1,yy + d1 z1,xx + d2 z1,y + d3 z1,x + d4 z2 + d5 z1 = 0, z1,xxx + e1 z1,xx + e2 z1,y + e3 z1,x + e4 z2 + e5 z1 = 0.

Appendix C Algebra of Monomials

In this appendix various properties of semigroups of monomials are described. Details and proofs may be found in the article by Janet [83]; see also Castro and Moreno [27]. The notation applied here is independent from the rest of the book. Let x1 , x2 , . . . , xn be n indeterminates equipped with the lex ordering αn 1 x1 < x2 < . . . < xn . Monomials in the xi are denoted by m = xα 1 . . . xn n with (α1 , . . . , αn ) ∈ N or, if several monomials are to be distinguished, by αi

αi

mi = x1 1 . . . xnn . Because the product of two monomials is again a monomial, and the product is commutative and associative, they form a commutative semigroup under multiplication. Because (1, . . . , 1) is the identity element, they even form a monoid. The connection of the results described in this appendix with partial differential equations is established by the identification α1 +...+αn of the degree vector (α1 , . . . , αn ) with the partial derivative ∂α1 n . ∂x1 . . . ∂xα n Let M be a finite set of such monomials. They generate an ideal in this monoid. This ideal should not be mistaken for an ideal in the polynomial ring k[x1 , . . . , xn ] with k some number field. The totality of all monomials of the monoid decomposes into those which are contained in the ideal generated by M and those which are not. For the applications in the main part of this book a unique representation of the members of both subsets is required. To this end for both subsets a decomposition into classes will be obtained. The classes are defined as the entirety of all multiples of any monomial of a complete set with respect its multipliers, i.e., a certain subset of its variables. It is the main subject of this appendix to define a procedure for constructing such a decomposition into classes for any given set of monomials. The finiteness of various algorithms dealing with monomials is based on the following lemma which is proved in Janet’s article [83], pages 69-70. Lemma C.1 A set of monomials such that none of its elements is a multiple of any other monomial is necessarily finite. Let M = {m1 , m2 , . . .} be a finite set of monomials and a1 < . . . < ak the degrees in xn which do actually occur in M. Mα ⊂ M denotes those monomials in M which occur for a fixed value of α ∈ {a1 , . . . , ak }. The first step toward the decomposition mentioned above is the subdivision of variables

383

384 and monomials as described in the following definitions. Definition C.1 (Multipliers and non-multipliers) The leading variable xn is called a multiplier for mi ∈ M if αni = ak ; otherwise it is called a nonmultiplier. If n > 1, the same procedure is applied to the monomials of Maj , j = 1, . . . , k, w.r.t. n − 1 variables x1 , . . . , xn−1 . Definition C.2 (Classes) The totality of all multiples of a monomial mi ∈ M with respect to its multipliers is called the class defined by this monomial and is denoted by mi . The union of the classes of all monomials in M is denoted by M. It should be noted that this definition of multipliers is only meaningful with respect to a fixed ordering for the variables. In general there will be monomials in the ideal generated by M which do not belong to any class at all. Example C.1 Consider the following set M comprising four monomials in three variables. M onomial M ultipliers M onomial M ultipliers x21 x22 x33 x31 x33

x1 , x2 , x3 x1 , x3

x31 x2 x3 x2 x3

x1 , x2 x2

The monomial x1 x2 x3 obviously may not be represented as the product of any monomial in M and some of its multipliers; and therefore does not belong to any class. The special sets of monomials which have the property that any monomial in the ideal generated by it belongs at least to a single class are defined next. Definition C.3 (Complete system) A set of monomials M is called complete if any monomial in the ideal generated by M belongs to at least one of its classes. Due to this property of complete sets it is desirable to have a procedure which determines for any given set of monomials an equivalent complete set, i.e., a complete set generating the same ideal as the given one. The following three theorems due to Janet are the prerequisite for such an algorithm. Theorem C.1

A set of monomials M is complete if and only if

i) The sets Ma1 , . . . , Mak are complete with respect to x1 , . . . , xn−1 . ii) For any monomial m ∈ M there holds m/xα n ∈ Mα → xn · m ∈ Mα+1 for α < ak . Theorem C.2 Let M be a set of monomials. If the product of any monomial in M with any of its non-multipliers is contained in M, then M is complete.

Algebra of Monomials

385

Theorem C.3 Any finite set of monomials M may be extended to a complete set generating the same ideal as M. The proof of these theorems may be found in Janet [83], page 74 ff. An obvious consequence of Theorem C.1 is the constraint ai+1 = ai + 1 for 1 ≤ i ≤ k −1 for complete sets. Based on these results the following algorithm Complete may be designed which is also due to Janet. The notation is the same as above. Algorithm C.1 Complete(M). Given a set M = {m1 , m2 , . . .} of monomials, the complete set corresponding to M is returned. S1 : Find Multipliers & Non-Multipliers. Determine multipliers and nonmultipliers for all monomials in M. S2 : Products with Non-Multipliers. Generate the set M0 of monomials which are the product of an mi w.r.t. any one of its non-multipliers. S3 : In Classes? Remove all those elements from M0 which are contained ¯ in M. S4 : Termination? If M0 is empty return M, else set M := M ∪ M0 and goto S1. Example C.2 If this algorithm is applied to example C.1, the following complete set of monomials is obtained. M onomial M ultipliers M onomial M ultipliers x21 x22 x33 x1 x22 x33 x22 x33 x31 x2 x33 x21 x2 x33 x1 x2 x33 x2 x33 x31 x33

x1 , x2 , x3 x2 , x3 x2 , x3 x1 , x3 x3 x3 x3 x3 , x1

x31 x2 x23 x21 x2 x23 x1 x2 x23 x2 x23 x31 x2 x3 x21 x2 x3 x1 x2 x3 x2 x3

x1 , x2 x2 x2 x2 x2 x2 x2 x2

Complete sets of monomials have several properties that make them important tools for analyzing sets of pde’s. They are collected in the following theorem. Theorem C.4 ties.

A complete set of monomials M has the following proper-

i) Any monomial in the ideal generated by M belongs exactly to a single of its classes. ii) The product of any monomial in M by one of its non-multipliers is equal to the product of some other monomial by its multipliers.

386 Let M denote a complete set of monomials. To obtain a description of the monomials which are not contained in the ideal generated by M, a new set of monomials, the complementary set N , will be constructed as follows. If an > 0 then xα n ∈ N for 0 ≤ α < a1 with multipliers x1 , x2 , . . . xn−1 . For 1 ≤ i ≤ k, consider Mai . Let b1 < b2 < . . . bj be the exponents of xn−1 which do occur in Mai . If b1 > 0 then the monomials xani xβn−1 ∈ N for 0 ≤ β < b1 . Its multipliers are x1 , x2 . . . xn−2 if i < k, and in addition xn if i = k. This proceeding is applied recursively all the way down until n = 1 is reached.

Appendix D Loewy Decompositions of Kamke’s Collection

In this appendix the types of Loewy decompositions of linear second and third order equations in x and y(x) with coefficients in Q from Kamke’s collection are listed. Excluded are equations containing undetermined functions, more than two parameters, or a parameter in an exponent. For these equations usually a more sophisticated analysis is required in order to determine the constraints leading possibly to different decomposition types. If there are parameters involved, the given decomposition applies to unconstrained parameter values. Any relations between them may change the decomposition type. For inhomogeneous equations only the left hand side is taken into account. From the given nontrivial decompositions fundamental systems may be obtained by integration according to the formulas on page 31. Equations of Second Order. These are the equations of Chapter 2 of Kamke’s collection. The tabulation below is organized by decomposition type as described in Corollary 2.3. For types L2k , k = 2, 3 or 4 the numbers of the corresponding equations are listed. For irreducible equations of type L21 a further distinction is made according to the type of Galois group of its rational normal form as discussed in Chapter 2 on page 35. If the Galois group is SL2 , the equivalence to an equation defining a special function is given whenever possible. The following equations defining special functions w(z) are taken into account.  2  w00 + z1 w0 − 1 − ν 2 w = 0, z (α + β + 1)z − γ 0 αβ 00 Hypergeometrical : w + w + w = 0. z(z − 1) z(z − 1) Bessel :

Legendre : w00 +

2z ν(ν + 1) w0 − 2 )w = 0. z −1 z −1 2

Weber : w00 − zw0 − kw = 0.  1 k 4m2 − 1  Whittaker : w00 − − + w = 0. 4 z 4z 2  a 0 b Confluent Hypergeometrical : w00 − 1 − w − w = 0. z z

387

388 L21 : Galois group SL2 , equivalent to Bessel equation. 2.10, 2.14, 2.15, 2.86, 2.94, 2.95, 2.102, 2.103, 2.104, 2.105, 2.106, 2.123, 2.130, 2.149, 2.155, 2.161, 2.162, 2.164, 2.165, 2.167, 2.169, 2.170, 2.172, 2.173, 2.180, 2.185, 2.189, 2.197, 2.200, 2.206, 2.216, 2.272, 2.274, 2.347. L21 : Galois group SL2 , equivalent to Hypergeometrical equation. 2.12, 2.252, 2.258, 2.265, 2.287, 2.291, 2.293, 2.294, 2.406. L21 : Galois group SL2 , equivalent to Legendre equation. 2.226, 2.231, 2.239, 2.240, 2.241, 2.244, 2.245, 2.249, 2.256, 2.269, 2.311, 2.313. L21 : Galois group SL2 , equivalent to Weber equation. 2.42, 2.44, 2.46, 2.87, 2.131, 2.132, 2.139. L21 : Galois group SL2 , equivalent to Whittaker equation. 2.16, 2.37, 2.114, 2.154, 2.195, 2.273. L21 : Galois group SL2 , equivalent to Confluent Hypergeometrical equation. 2.12, 2.92, 2.96, 2.107, 2.110, 2.113, 2.134, 2.138, 2.190. L21 : Imprimitive Galois group Dm with m > 24, or D. 2.130, 2.135, 2.222, 2.288, 2.289. L21 : Imprimitive Galois group D3 . 2.290, 2.292. L22 : 2.11, 2.13, 2.39, 2.40, 2.41, 2.43, 2.45, 2.50, 2.56, 2.57, 2.58, 2.59, 2.93, 2.109, 2.121, 2.122, 2.133, 2.136, 2.141, 2.158, 2.166, 2.181, 2.182, 2.192, 2.194, 2.196, 2.198, 2.199, 2.208, 2.209, 2.223, 2.225, 2.230, 2.234, 2.237, 2.238, 2.242, 2.246, 2.251, 2.253, 2.254, 2.257, 2.262, 2.263, 2.264, 2.267, 2.270, 2.271, 2.280, 2.299, 2.300, 2.304, 2.308, 2.310, 2.324, 2.326, 2.319, 2.321, 2.328, 2.332, 2.334, 2.336, 2.338, 2.345, 2.347a, 2.351, 2.355, 2.358, 2.365, 2.378, 2.379, 2.384, 2.386, 2.387, 2.399. L23 : 2.2,... 2.7, 2.35, 2.47, 2.49, 2.51, 2.53, 2.91, 2.98, 2.100, 2.101, 2.111, 2.112, 2.115, 2.119, 2.125, 2.126, 2.129, 2.148, 2.150, 2.151, 2.152, 2.160, 2.165a, 2.176, 2.179, 2.191, 2.193, 2.201, 2.202, 2.203, 2.204, 2.211, 2.243, 2.247, 2.255, 2.276, 2.277, 2.279, 2.282, 2.320, 2.331, 2.337, 2.342, 2.350, 2.353, 2.354, 2.361, 2.370, 2.380, 2.400, 2.401, 2.404, 2.405. L24 , Galois group Z1 : 2.1, 2.146, 2.147, 2.174, 2.183, 2.184, 2.229, 2.250, 2.281, 2.312, 2.322, 2.323, 2.340, 2.346, 2.366. L24 , Galois group Z2 : 2.159, 2.168, 2.175, 2.186, 2.227, 2.229, 2.266, 2.284, 2.286, 2.391. In the subsequent listing the number of the equation and its decomposition type is given. If the equation is irreducible and it is equivalent to any of the known equations defining a special function, the transformation to this equation is given; as usual D = d . dx

Loewy Decompositions

389

1
y = 0. Type L2 . 2.1: y 00 = Lclm D − C + 4 x 00 2.2,...,2.5: y + y = Lclm(D − cot x, D + tan x)y = 0. Type L23 . 2.6: y 00 − y = Lclm(D + 1, D − 1)y = 0. Type L23 . √ √ 2.7: y 00 − 2y = Lclm(D + 2, D − 2)y = 0. Type L23 . 2.8: y 00 + a2 y = Lclm(D + ia, D − ia)y = 0. Type L23 . √ √ 2.9: y 00 + ay = Lclm(D + i a, D − i a)y = 0. Type L23 . 2.11: y 00 − (x2 + 1)y = (D + x)(D − x)y = 0. Type L22 . 2.13: y 00 − (a2 x2 + a)y = Lclm(D + ax, D − ax)y = 0. Type L22 . q q   2.35: y 00 + ay 0 + by = Lclm D + 12 a + 14 a2 − b, D + 12 a − 14 a2 − b y = 0. Type L23 . 2.39: y 00 + xy 0 + y = D(D + x)y = 0. Type L22 . 1
D − 1
y = 0. Type L2 . 2.40: y 00 + xy 0 − y = D + x + x 2 x 0
0
p p 2.41: y 00 + xy 0 + (n + 1)y = D + pnn D + x − pnn y = 0. Type L22 ; pn monic polynomial of degree n; p1 = x, p2 = x2 −1, p3 = x3 −3x, p4 = x4 −6x2 +3. 2.42: y 00 + xy 0 − ny = 0. Type L21 . Weber equation, k = −n, x = iz, y = w. 1 + 1
D − 1 − 1
y = 0. 2.43: y 00 − xy 0 + 2y = D − x + x + 1 x−1 x+1 x−1 Type L22 . 2.44: y 00 − xy 0 − ay = 0. Type L21 . Weber equation. 2.45: y 00 − xy 0 + (x − 1)y = (D − x + 1)(D − 1)y = 0. Type L22 . 2.46: y 00 − 2xy 0 + ay = 0. Type L21 . Weber equation, k = − 21 a, x = √12 z, y = w. 1 , D + 2x
y = 0. Type L2 . 2.47: y 00 + 4xy 0 + (4x2 + 2)y = Lclm D + 2x − x 3 00 0 2 2 2 2.49: y − 4xy + (4x − 1)y = Lclm(D − 2x − θ) = exp x , θ + 1 = 0. Type L23 . 2.50: y 00 − 4xy 0 + (4x2 − 2)y = (D − 2x)(D − 2x)y = 0. Type L22 . 2.51: y 00 − 4xy 0 + (4x2 − 3)y = Lclm(D − 2x + 1, D − 2x − 1)y = 0. Type L23 . 2.53: y 00 + 2axy 0 + a2 x2 y = Lclm(D + ax − θ) = 0, θ2 − a = 0. Type L23 . 1
D − 1
y = 0. Type L2 . 2.56: y 00 − x2 y 0 + xy = D − x2 + x 2 x 2.57: y 00 − x2 y 0 − (x2 + 2x + 1)y = (D + 1)(D − x2 − 1)y = 0. Type L22 . 2.58: y 00 − (x3 + x2 )y 0 + (x5 − 2x)y = (D − x3 )(D − x2 )y = 0. Type L22 . 1
D − 1
y = 0. Type L2 . 2.59: y 00 + x4 y 0 − x3 y = D + x4 + x 2 x 2.86: y 00 + 49 xy = 0. Type L21 . Bessel equation, ν = 31 , x = z 2/3 , y = z 1/3 w. 2.87: y 00 − 41 (x2 + a)y = 0. Type L21 . Weber equation, k = − 14 (a + 1), x = z, y = exp(− 41 z 2 )w.

390 2.91: y 00 + y = Lclm(D − θ)y = 0, θ2 + 1 = 0. Type L23 . 1 y 0 = D + 1
Dy = 0. Type L2 . 2.93: y 00 + x 2 x 1 a z 2 , y = w. 00 0 2 2.94: y + x y + x y = 0. Type L1 . Bessel equation, ν = 0, x = 4a 1 y 0 + λy = 0. Type L2 . Bessel equation, ν = 0, x = √z , y = w. 2.95: y 00 + x 1 λ
1 c 00 0 2 2.96: y + x y + 1+ x y = 0. Type L1 . Confluent Hypergeometric equation, a = 1, b = 12 (1 + ic), x = 21 iz, y = exp (− 12 z)w. 1 y 0 − ax2 y = Lclm(D − θx) = 0, θ2 − a = 0. Type L2 . 2.98: y 00 − x 3
2 1 1 00 0 2.100: y + x y − y = Lclm D + 1 + x , D − 1 + x y = 0. Type L23 . 2 y 0 + ay = Lclm D − θ + 1
= 0, θ2 + a = 0. Type L2 . 2.101: y 00 + x 3 x β 0 2 2.104: y 00 + α x y + x y = 0. Type L1 . Bessel equation, ν = α − 1, 2
(1−α)/2 2 w. x= z ,y= z 4β 4β a y 0 + by = 0. Type L2 . Bessel equation, ν = 1 − a , 2.105: y 00 + x 1 2  ν z z x = −√ , y = − √ w. b b −a, a 00 0 α−1 2.106: y + x y + bx y. Type L21 . Bessel equation, ν = 11 + α     (α + 1)2 z 2 1/(α+1) (α + 1)2 z 2 (1−a)/(2α+2) x= ,y= w. 4b 4b

1 y = D + 1 D − 1 − 1 y = 0. Type L2 . 2.109: y 00 − y 0 − x 2 x x

0 1 1 1 00 2.111: y − 1 + x y + x y = Lclm D − 1, D − 1 + x + 1 y = 0. Type L23 . 1
y 0 − 2 − 2
y = Lclm D + 1 − 1 , D − 2
y = 0. 2.112: y 00 − 1 + x x x + 13 2 Type L3 . 2 )y 0 + 2 − 3
y = Lclm D − 1 + 1 , D − 2 + 1
y = 0. 2.115: y 00 − (3 − x x x x Type L23 .
2(ax + b) 0 a(ax + 2b) 2.119: y 00 − y + y = Lclm D − a + 2b x+ 1 , D − a y = 0. x x Type L23 . 1
y = D − x + 1 + 1 )(D − 1
y = 0. Type L2 . 2.121: y 00 − (x − 1)y 0 + 1 − x 2 x x
0
2 1 00 2.122: y − x − 1 − x y − (x + 3)y = D + 1 + x (D − x)y = 0. Type L22 . 1
y 0 − 4x2 y = Lclm(D − 2αx, D − 2βx)y = 0 where 2.125: y 00 + 4x − x α + β + 2 = 0, αβ + 1 = 0. Type L23 . 1
y 0 + (ax4 + ax)y = Lclm D + ax2 − 2 , D + ax2
y = 0. 2.126: y 00 + 2ax2 − x x Type L23 . − 9 0 3x − 6 2.129: y 00 − 4x x−3 y + x−3 y =

Loewy Decompositions

391

12x2 − 28x + 50 y = 0. Type L23 . 4x3 − 42x2 + 150x − 183 − 1 y 0 + a y = 0. Type L2 . Weber equation, k = −2a, 2.131: y 00 − x 2x 1 2x x = z 2 , y = w. 5
x−3 1 −4 0 2 2 2.133: y 00 − 3x 2x − 1 y + 2x − 1 y = D − 2 + x − 1 (D − 1)y = 0. Type L2 . 2 + c y = 0. Type L2 . Confluent Hypergeometric equation, 2.134: y 00 − x4x 1
a = 2, b = 1 + 4c , x = z, y = z exp − 12 z w. 1 y 0 − x + 2 y = D + 1 + 1
D − 1
y = 0. Type L2 . 2.136: y 00 + x 2 2 x 2 4x 

Lclm D − 1, D − 3 −



1 y 0 − x + a y = 0. Type L2 . Weber equation, k = − 1 (a + 2) 2.139: y 00 + 2x 1 4 16x
x = z 2 , y = exp − 14 z 2 w. + bx y 0 + 3b y = D + 3
D + b
y = 0. Type L2 . 2.141: y 00 + 3a ax 2 ax x a   4 2 y = 0. Type L2 . 2.146: y 00 − 62 y = Lclm D − 55x +x 4 x x +C   6 3 y = 0. Type L2 . 2.147: y 00 − 122 y = Lclm D − 77x +x 4 x x +C
θ y = 0, θ2 − θ + a = 0. Type L2 . 2.148: y 00 + a2 y = Lclm D − x 3 x √ b y = 0. Type L2 . Bessel equation, ν = 1 − 4b, 2.149: y 00 + ax + 1 x2 2 z z x = 4a , y = √ w. 2 a
1
y = 0. Type L2 . 00 2.150: y + 1 − 22 y = Lclm D + θ − 1 + x 3 x + θ x √ √  
a a 2.151: y 00 − a + 22 y = Lclm D + a − x + 12 , D − a − x − 12 y = 0. x x x Type L23 .  
2 − 2x + 3θ y = 0. Type L23 . 2.152: y 00 + 1 − 62 y = Lclm D − θ + x 2 x x + 3θx − 3 ν(ν − 1)
2.153: y 00 + a − y = 0. Type L21 . Bessel equation, ν + 12 , 2 √x i z x = − √z , y = 1/4 w. a a   b(b − 1) 2.155: y 00 + axk−2 − y = 0. Type L21 . Bessel equation, 2 x   2 2 1/(2k) p 2 2 1/k ν 2 = 1 1 + 4b(b − 1), x = k4az , y = k4az w. k

2.158: y 00 + a2 y 0 − b2 + ab2 y = Lclm D + b + a2 , D − b y = 0. Type L22 . x x x 1 y 0 − y = Lclm D − 1 1
y = 0. Type L2 . 2.159: y 00 − x +x 4 2 x +C 1
1 y 0 + a y = Lclm D − θ − 2 y = 0, θ2 + a + 1 = 0. Type L2 . 2.160: y 00 + x 3 x 4 x2

392 1 y 0 − 1 + a
y = 0. Type L2 . Bessel equation, ν = 2√a, 2.161: y 00 + x 1 x x2 1 2 x = − 4 z , y = w. 1 y 0 + λ − ν 2
y. Type L2 . Bessel equation, x = √z , y = w. 2.164: y 00 + x 1 x2 λ 2
1 ν 2.165: y 00 + x y 0 + 4 x2 − 2 y = 0. Type L21 . Bessel equation, x √ x = z, y = w. 1 + a
y 0 − y = Lclm D − 1 + a , D − 1 + 1
y = 0. 2.165a: y 00 + x x x2 x+a x2 Type L23 . 1 y 0 + 1 y = D D − 1
y = 0. Type L2 . 2.166: y 00 − x 2 x x2
1 y 0 + axm−2 + b y = 0 Type L2 . Bessel equation, 2.167: y 00 − x 1 2  2 2 x1/m  2 2 1/m 4(1 − b) ν2 = , x = m4az , y = m4az w. m2
2 y 0 = Lclm D − 1 + 1 = 0. Type L2 . 2.168: y 00 + x 4 x x+C 2
2 y 0 + a − b y = 0. Type L2 . Bessel equation, ν 2 = 1 + 4b2 , 2.169: y 00 + x 1 x x2 √ 2 z , y = 2 aw . x = 4a z
2 00 0 2.170: y + x y + a + b2 y = 0. Type L21 . Bessel equation, ν 2 = 14 − b, x 1/4 z a w √ √ x= ,y= . a z 2 y 0 + 2 y = Lclm D + 1 − 1 − 1
y = 0. Type L2 . 2.174: y 00 − x 4 x x+C x2 4
2 y 0 − 4 y = Lclm D + 1 − 5x 2 2.175: y 00 − x x x5 + C y = 0. Type L4 . x2 2 y 0 + 1 + 2
y = Lclm D − i − 1 , D − i + 1
y = 0. Type L2 . 2.176: y 00 − x 3 x x x2

2 y 0 + a + 2 y = Lclm D − θ − 1 y = 0, θ2 + a = 0. Type L2 . 2.179: y 00 − x 3 x x2 2
3 y 0 + 1 + 1 − k y = f (x). Type L2 . Bessel equation, ν = 0, 2.180: y 00 + x 1 x2 w x = z, y = − z . 1 y 0 + 1 y = D + 2
D + 1 − 1
y = 0. Type L2 . 2.181: y 00 + 3x − 2 2 x x x2 x x2

3 y 0 + 4 y = D − 1 D − 2 y = 5 . Type L2 . 2.182: y 00 − x 2 x x x x2 3 y 0 − 5 y = Lclm D − 6x5 + 1
y = x2 log x. Type L2 . 2.183: y 00 − x 4 x x2 C + x6
4 y 0 + 6 y = Lclm D + 1 + 2 y = x2 − 1. Type L2 . 2.184: y 00 − x 4 C +x x x2
5 y 0 − 2x − 4 y = 0. Type L2 . Bessel equation, ν = 0, 2.185: y 00 + x 1 x2

Loewy Decompositions

393

4(3z)2/3 w . 9z 2 5 y 0 + 8 y = Lclm D − 2 − 2x
y = 0. Type L2 . 2.186: y 00 − x 4 x x2 + C x2
a y 0 + bxm−2 + c y = 0. Type L2 . Bessel equation, 2.189: y 00 + x 1 2 4x  2 2 1/m  2 2 (1−a)/(2m) p 1 (1 − a)2 − 4c, x = m z ν=m ,y= m z w. 4b 4b   1 , D − 1 + 1 y = 0. 1 + 2x 2.191: y 00 − y 0 − 22 y = Lclm D + 1 − x−2 x 2(x + 2) x Type L23 .

2.192: y 00 + 1 − 12 y 0 − 12 y = D − 12 (D + 1)y = 0. Type L22 . x x x 

0 1 3 − 3x2 + 18x + 36 , 1 9 00 2.193: y + 1 + x y + x − 2 y = Lclm D + 1 + x x x3 + 9x2 + 36x + 60  2x − 8 3 2 D+ x − 2 y = 0. Type L3 . x − 8x + 20 1
y 0 +( 3 − 1 )y = D+ 1 + 2
D+1− 1 − 1
y = 0. 2.194: y 00 + 1+ x x x2 x−3 x x−3 x 2 Type L2 . 1
y 0 + 1 − 1
y = D − 1 + 2
D − 1
y = 0. Type L2 . 2.196: y 00 − 1 − x 2 x x2 x x



2 y 0 − 3 + 2 y = D + 3 D − 1 − 1 y = 0. Type L2 . 2.198: y 00 − 1 − x 2 x x2 x x 4 4 4 00 0 2 2.199: y − (1 + x )y − 2 y = D(D − 1 − x )y = 0. Type L2 . x
0 1 00 2.201: y + 2 + x y − 42 y = x   2(x − 1) 1 + 2 y = 0. Type L2 . Lclm D − 2 3 3 3 ,D + 2 − x x+ 2 x − 2x + 2 2
y 0 + 2 + 2
y = Lclm D − 1 , D −2− 1
y = 0. Type L2 . 2.202: y 00 − 2+ x 3 x x2 x x
2 a 1 a 1 00 0 2.203: y + ay − 2 y = Lclm D − ax − 2 + x , D + a + ax + 2 + x y = 0. x Type L23 .
2.204: y 00 + a + 2b)y 0 + (ab + b2 − 22 y = x   1 , D + ab − 2b + 1 y = 0. Type L2 . Lclm D + a + b − ax 1+ 2 + x 3 x x


2 1 1 2.208: y 00 + xy 0 + 1 − 2 y = D + x − x D + x y = 0. Type L22 . x 2 2 x + 2 2 2
00 0 2 2.209: y + x y + x − 2 y = D D + x + x y = 0. Type L2 . x θ )y = 0, θ2 − θ + 1 = 0. 2.211: y 00 + 4xy 0 + (4x2 + 2 + 12 )y = Lclm(D + 2x − x x Type L23 . 2.223: y 00 + 2 x y 0 − 2 9 y = x +1 x + 1   2x x 1 D − 2x + 1 y = 0. Type L2 . D+ 2 3 + 2 +x 2 x x + x +1 x2 + 3 x = − 12 (3z)2/3 , y = −

4

4

394 x y 0 + 1 y = D + 1 − x
D − 1
y = 0. Type L2 . 2 x x2 + 1 x x2 + 1 x2 + 1 1  1+ 2  2x 2 1 2 x 2.227: y 00 − 2 y0 + 2 y = Lclm D− x − 1 y = 0. Type L4 . x +1 x +1 C +x− x  4x 2 1 2x 00 0 2.229: y + 2 y+ 2 y = Lclm D− C + x + 2 y = 2 cos2 x − 2x . x +1 x +1 x +1 x +1 Type L24 .    (a − 2)x D+ 2 y = 0. 2.230: y 00 + 2ax y 0 + a2− 1 y = D + 22x x +1 x +1 x +1 x +1 2 Type L2 .  1    1 1 1 0 2 2 2 2 2 2.234: y 00 + x + + y = D + + 1 x−1 x + 1 x − 1 Dy = 0. Type L2 . 1 + 1
y 0 = D + 1 + 1
Dy = 0. Type L2 . 2.237: y 00 + x + 2 1 x−1 x+1 x−1
2.238: y 00 + 22x y 0 = D + 22x Dy = 0. Type L22 . x −1 x −1 x 2 − 1
D + 1 − 2
y = 0. 3x + 1 00 − x + 1y = D + x − 2.242: y − 2 1 x+1 x+1 x −1 Type L22 . 2 1 + 1
y = 0, θ2 +1 = 0. 2.243: y 00 + 24x + x2 + 1 y = Lclm D−θ+ x + 1 x−1 x −1 x −1 Type L23 . 1 + 1 )y 0 − a(a − 1) ( 1 − 1 )y = 2.247: y 00 + a( x + 1 x−1 2 x+1 x−1 a − 1 , D + a − 1 )y = 0. Type L2 . Lclm(D + x 3 +1 x−1
0 1 4 4 4
= 4 00 2.250: y + x + a + x − a y − a x + a + x − a   12C(a2 + x2 )x + 9a4 + 54a2 x2 + 9x4 Lclm D − y = 0. Type L24 . 4 2 2 4 4 2 2 4 C(a + 2a x − x ) + 3x(3a − 2a x − x ) 2 − 1
y = D − 2 + 1 + 1
D − 1
y = 0. 2.251: y 00 − x + 1 x x+1 x−1 x x−1 Type L22 . 2 + 1
y = D + 1 + 1
D − 1 + 1
y = 0. Type L2 . 2.253: y 00 + x + 2 1 x x+1 x x+1 x

2 3 3 2.254: y 00 + 1 − x + 2 y 0 − 6 − x + 2 + x − 1 y = 2 + 1
D − 2 + 1
y = 0. Type L2 . D+3− x+ 2 2 x−1 x−1

1 − 1 y0 − 2 2 2.255: y 00 + a x − 1 x x−1 − x y =   a − 1 − a + 1, D − 2x + 2a − 1 Lclm D + x y = 0. 2 x −1 x + ax − 2 + a(a − 1)/2 2.225: y 00 +

Type L23 . (2x − 1 0 ν(ν + 1) y − y = 0. Type L21 . Legendre equation, x(x − 1) x(x − 1) z x= 1− 2 , y = w.

2.256: y 00 +

Loewy Decompositions

395

(a + 1)x + b 0 b
b+1 2 y = D+ a+ x − 1 − x Dy = 0. Type L2 . x(x − 1)
2 1 1 − (D+1)y = 0. 2.262: y 00 + x + x −2 1 y 0 − x + 2 2 y = D− x + 1 x + 2x + 1 (x − 1) (x − 1) Type L22 . 1 − 1 y0 + y= 2.263: y 00 + 3x (x2 + 3x)2 (x2 + 3x)    4 7 +3 3 D + 2x − x3 y = 0. Type L22 . D+ x+ 2 3 x + 3x 2 2.264: y 00 + x2 + x + 1 y 0 − 2 2x + 3 y = x + 3x + 4 x + 3x + 1    2x + 1 2x + 3 D+ 2 − 3 D − 22x + 1 y = 0. Type L22 . x + x + 3 x + 3x + 4 x +x+3 2x − 3 1 2.265: y 00 − y0 + y = 0. Type L21 . (x − 1)(x − 3) (x − 1)(x√ − 3) √ Hypergeometric equation, α = − 21 (3 + 5), β = − 12 (3 − 5). γ = −1, x = z + 1, y = w.   4(x − 1)3 ) 1 y0 − 3 1 D − 2.266: y 00 − x − y = Lclm − 2 x − 2 y = 0. (x − 2)2 C + (x − 2)4 Type L24 . 1 ν + x2 − 52 0 2x − 12 2
D − 1 + 1 − ν
y = 0. 2.267: y 00 − 2 y + y= D+ x 2 2 2x 2x2 x x Type L22 . (2ν + 5)x − 2ν − 3 0 2.269: y 00 + y + ν + 1 y = 0. Type L21 . Legendre 2x(x − 1) 2x(x − 1) x + 1 equation, inverse of z = √ and w = x(ν+1)/2 y. 2 x 2 5x + 21/2x + 4 0 6x2 + 17/2x + 4 y + y= 2.270: y 00 + 2 x + 3x +2 x2 +   3x + 2 2 4 2 D + 4x 2+ 11x + 4 D + 3 − x + 2 y = 0. Type L2 . 2x + 6x + 4 1
D − 1
y = 0. Type L2 . 2.271: y 00 + 1 2 y = D + 2x 2 2x 4x
2.272: y 00 + a2 + 1 4 y = 0. Type L21 . Bessel equation, ν = 0, 4x z , y = z
1/2 w. x= a a 2 1 00 2.274: y + x y 0 + x − 2ν y = 0. Type L21 . Bessel equation, x = z 2 , y = w. 4x √ x 2 1 4x + 1 y = Lclm D + 1 + 1 , D − 1 + 1
y = xe . 2.276: y 00 + x y 0 − x 2x 2x 4x2 Type L23 . √ √ 1 y 0 − ax2 + 1 y = Lclm D + a − 1 , D − a + 1
y = 0. 2.277: y 00 + x 2 2x 2 2x 4x2 Type L23 . 5 y 0 − 1 y = Lclm D − θ
y = 0, θ2 + 1 θ − 1 = 0. Type L2 . 2.279: y 00 + 4x 3 x 4 4 4x2 2.257: y 00 +

396 2 3 2 y 0 − x + 3x + 4 y = D + 1 + 5
D − 1 − 1
y = 0. Type L2 . 2.280: y 00 + x 2 2 2x 2x x
2 − 1 y = Lclm D − 2x − 1 − 1 2.281: y 00 − 2x x− 1 y 0 − 4x − 4x 2x C + x y = 0. 4x2 2 Type L4 .
2 , D + 1 − 3
y = 0. 2.282: y 00 + xy 0 + 14 x2 + 12 − 6 y = Lclm D + 12 x + x x x Type L23 . 12 2.284: y 00 − 2x 2+ 1 y 0 − y= (2x + 1)2   4x3 + 6x2 + 3x + 12 Lclm D − 1 1 + y = 0. Type L24 . 4 3 3 2 1 x+ 2 C + x + 2x + 2 x + 2 x 9 2.286: y 00 + 3x 3− 1 y 0 − y= (3x − 1)2   2 2x − 3 1 Lclm D − y = 0. Type L24 . + C + x2 − 23 x x − 13

3(2x − 1) 0 20 y − y = 0. Type L21 . Hypergeometric 9x(x − 1) 9x(x − 1) equation, α = 31 , β = − 20 3 . 3 2.290: y 00 + 27x y0 − y = 0. Type L21 . Galois group D3 . 27x2 + 4 27x2 + 4
2 2.299: y 00 + 22a2 x y 0 = D + ax a+ 1 + ax a− 1 Dy = 0. Type L22 . a x −1 2 2 1
D− 1
y = 0. 00 y = D+ ax a+ 1 + ax a− 1 + x 2.300: y + 22a2 x y 0 − 2 2a 2 x a x −1 a x −1 Type L22 . 3 y = D − 1 , D + 1 + 1
y = 0. Type L2 . 2.304: y 00 + 12 y 0 − 2x + 2 3 x x x2 x x 1 , D + 1 + 1 − 1
y = 0. 2.307: y 00 + x +2 1 y 0 − 23 y = Lclm D − x + x x2 1 x+1 x x 2 Type L2 . 3 1 y 0 + 1 y = D D − 1
y = log (x) . Type L2 . 2.308: y 00 − x 2 x x2 x3

3 y 0 + 1 y = D + 2 D + 1 y = 1. Type L2 . 2.310: y 00 + x 2 x x x2 2 ν(ν + 1)x 2.311: y 00 + 2x 2 + 1 y 0 − y = 0. Type L21 . Legendre equation, 2 x(x + 1) x + 1 √ x = z 2 − 1, y = w.   2 2 2.312: y 00 + 2x3 − 2 y 0 − 2 2 y = Lclm D − 3x 3 + 2 2x y = 0. x +x x +1 C +x x + 1) 2 Type L4 . 1
D − x − 2
y = 0. 2.319: y 00 − 3 1 y 0 − 2 6 y = D + 22x + x x +x x +1 x +1 x2 + 1 x 2 Type L2 . 2.287: y 00 +

3 2 − 2x − 1 y 0 + x4 + 4x + 2 y = 2.320: y 00 − x + 3x 3 x − 2x x3 − 2x

Loewy Decompositions

397

2 , D − 1
y = 0. Type L2 . Lclm D − 1 − x 3 x−1 + 1 y 0 + 2x + 1 y = D − 1
D − 1
= 0. Type L2 . 2.321: y 00 − 2x 2 x x+1 x2 + x x3 + x2
+ 4 y 0 + 6x + 2 y = Lclm D − 1 + 3x + 2 y = 0. 2.322: y 00 + 6x x+C x(x + 1) x2 + x x3 + x2 2 Type L4 .   − 1 y 0 − 2x + 2 y = Lclm D − 3x2 − 6x + 3 + 2 y = 0. 2.323: y 00 + 2x 2 3 2 3 2 x x −x x −x C + x − 3x + 3x Type L24 . − 4 y 0 + 9x − 6 y = D − 1 − 1
D − 3
y = 0. Type L2 . 2.324: y 00 − 5x 2 2 x x−1 x x −x x3 − x2

1 y0 + 1 D + 1 − 1 y = 0. 1 2.326: y 00 + x + y= D+ x 1 x+1 x x3 + 2x2 + x Type L22 .   2 1 2− 2 + 1 1
1
2.328: y 00 − x 2 x − 1 (x − 1) y = D+ x − x − 1 D− x + x − 1 y = 0. Type L22 .  1 1 3  1 1
y 0 − 2 8 8 − 4 y = − − 2.331: y 00 + x − 2 x x − 2 x x2 1 1 1 2 2 Lclm(D − x − 2 + 2x , D − 2x )y = 0. Type L3 . 1 y 0 + 3x + 1 y = D − 1 + 1
D − 1
y = 0. 2.332: y 00 − x + 1 x + 1 2x 2x 4x3 + 4x2 2 Type L2 . ν(ν + 1) 2.333: y 0 + 3x − 1 y 0 − y = 0. Type L21 . Legendre equation, 2x(x −√1) 4x2 x = 2z 2 − 1 + 2z z 2 − 1, y = w. − 4 y 0 + 9x − 6 y = D + 1 − 1
D − 3
y = 0. Type L2 . 2.334: y 00 − 5x 2 2 x x−1 x x −x x3 − x2 3x − 4 2.336: y 00 − y= (x − 1)(2x − 1)2    1 1 1 − 2 1 + 2 D+ x− D − y = 0. Type L22 . 1 1 1 x− x−1 x− 2 2 a−b 2.337: y 00 + 3x + a + 2b y 0 + y= 2(x + a)(x + b) 4(x + a)2 (x + b)2  1 a−b ,D − y = 0. Type L23 . Lclm D + 2(x + a) 2(x + a)(x + b) − 3 y0 − 3 2.338: y 00 − 18x y= 9x2 − 2x 9x3 − 2x2   3 2x − 13 1+ 3 2 Lclm D + x 2 2 y = 0. 1 1 , D + 2x + x− 9 x − 3 x + 54 Type L23 . + b y 0 + 2ax + 6b y = Lclm D − 1 + ax + b
y = 0. 2.340: y 00 − 2ax 2 x + C ax2 + bx ax + bx ax3 + bx2 2 Type L4 .

398 √ √   1 + i a , D + 1 − i a y = 0. Type L2 . 2.342: y 00 + a4 y = Lclm D + x 3 x x x2 x2

1 D − 1 + 1 y = 0. Type L2 . 2.345: y 00 + 13 y 0 − 24 y = D + x 2 x x3 x x
1 y = Lclm D − 1 − 1 2 2.346: y 00 − 22 y 0 + 2x + x + C y = 0. Type L4 . x x4 x2 1 y 0 + 1 y = 0. Type L2 . Bessel equation, ν = 0, x = 1 , y = w. 2.347: y 00 + x 1 z x4

1 y 0 + x − 1 y = D + 1 + 1 D − 1 y = 0. Type L2 . 2.347a: y 00 + x 2 x x2 x4 x2
2 2 y 0 + a y = Lclm D − θ y = 0, θ2 + a2 = 0. Type L2 . 2.350: y 00 + x 3 x4 x2

2 2 D + 1 y = 0. Type L2 . 2.351: y 00 + 2x 3+ 1 y 0 − 14 y = D + x 2 x x x3  
3 2 − 1 y 0 + 2 y = D + 1 − 4x + 4x 2.353: y 00 − x x x4 + 2x2 − 1 x3 x4  1 3 2 2 23 x − 7x 3
y = 0. Type L2 . 7 7 D + 21 + x− +x + x+ 2 4 2 1 1 x + 3x + 3
2 − 1 y0 + 1 y = 2.354: y 00 − x x3 x4    1 2 4 2x 2− D− x + 3 + 2 1 D+ x y = 0. Type L22 . x x − x2 − 1 5

5

 2 − 23 1 y0 − x y = 3 − 2.355: y 00 + + x+ x 1 x −x+1 x3 +  1  2 1 x 2x − 1 1 − D− 3 y = 0. Type L22 . D+ 2 + x −x+1 x+1 x x +1   1 1 2 2 1 D − 1
y = 0. 2 2 2.358: y 00 − x3 − 2 y 0 − x4 − 2 y = D + x + + − x 1 x−1 x x −x x −x Type L22 . 3 ν(ν + 1) 2.359: y 00 + 2 2x y0 + 2 2 y = 0. Type L21 . Legendre equation, 2 x (x − 1) x (x − 1) x = z1 , y = w. 

4 3x 2

a2 (x2 − 1) + a(3x2 − 1) 2.361: y 00 − 22x y 0 − y= x −1 x4 − x2   a,D − a + 1 − 2x Lclm D + x y = 0. Type L23 . 2 3 1 x x − (a + 2 )/(a + 2 ) 1 − 1 + x
y = 0. Type L2 2.365: y 00 + 2 a 2 y = D D − x + 2 1 x − 1 x2 + 1 (x + 1)
1 + x 2.366: y 00 + 22x y 0 + 2 1 2 y = Lclm D − x + y = 0. C x +1 (x + 1) x2 + 1 Type L24 2 2.370: y 00 + 22x y 0 − 2 a 2 y = x −1 (x − 1)   a 1 1
, D − a 1 − 1
y = 0. Type L2 . Lclm D + 2 x + 1 − x − 3 1 2 x+1 x−1

Loewy Decompositions 2.378: y 00 +

399

2(x + 1) 0 2(x2 − x − 1) 3
D+( 1 − 2
y = 0. y = D+ x − y− 2 2 1 x−1 x x(x − 1) x (x − 1)

Type L22 . 12 2.379: y 00 − y= 2 2 (x + 1) (x + 2x + 3)   4 4 D− D + y = 0. Type L22 . (x + 1)(x2 + 2x + 3) (x + 1)(x2 + 2x + 3) m(m − 1)a2 m m − 1 , D+ m − 1 − m
y = 0. 2.380: y 00 + 2 2 y = Lclm D+ x − a − x x−a x x (x − a) 2 Type L3 .   (ax − b)2 − 6bx − x2 (ax − b)2 − 2bx − x2 2.384: y 00 − y = D − 4x2 (2a + 1)x3 − 2bx2   +1 − b − 1 D + a 2x y = 0. Type L22 . x − b/(a + 1) 2x2 18 2.386: y 00 − y= 2 2 (2x + 1) (x +  x + 1)   D − 22x + 1 + 2 1 D + 22x + 1 − 2 1 y = 0. Type L22 . x +x+1 x+ 2 x +x+1 x+ 2  3 x + 12  x + 12  2.387: y 00 − 2 4 D− 2 y = 0. 2y = D + 2 (x + x + 1) x +x+1 x +x+1 Type L22 .   3 3 1 3 1 4 4 ,D − 4 − 4 2.390: y 00 + 2 16 2 y = Lclm D − x − − x x 1 x − 1 y = 0. x (x − 1) 2 Type L3 . 2 2 + 5y = 2.391: y 00 − 7ax3 + 5 y 0 + 15ax 4 ax + x ax + x2   (8ax3 (ax2 + 1))/(2ax2 + 1)2 6ax2 + 1 y = 0. Type L24 . Lclm D − + C + (2ax4 )/(2ax2 + 1) x(2ax2 + 1 1 − 1
D − 1
y = 0. Type L2 . 2.397: y 00 + 14 y 0 − 15 y = D − x 2 x x x x4 2 12(x + 1) 2.399: y 00 − 3x2 + 1 y 0 + y= x −1 (x − 1)2 (x + 35 )2 1 1 3 1 1 − 2 + 2
D − 2 − 2
y = 0. Type L2 . D− x+ 2 1 x−1 x+ 5 x−1 x+ 5 3 3 √ √
1 y 0 + a y = Lclm D − 2 + a , D − 2 − a y = 0. Type L2 . 2.400: y 00 − x 3 x x x6 x3 x3 2 2 a m(m − 1) 2.401: y 00 + 3x 3+ a y 0 − y= x x6  a(m − 1) Lclm D − , D + am y = 0. Type L23 . x3 x3 2 2 1y = 2.404: y 00 + 2x 3+ 1 y 0 − 2x − x 4x6
1 + 1 , D + 1 y = 0. Type L2 . Lclm D + x 3 2x3 2x3

400 2 4 2 + 1y = 2.405: y 00 + 2x 3+ 1 y 0 − 8x + 10x x 4x6
1 − 1 , D − 2 − 1 y = 0. Type L2 . Lclm D − x 3 x 2x3 2x3 x 1 1 1
2 2.406: y 00 + 27 16 (x3 − 1)2 y = Lclm D + x + 2x3 , D + 2x3 y = 0. Type L1 .

Equations of Third Order. These equations are from Chapter 3 of Kamke’s collection. The tabulation below is organized by decomposition type L3k , k = 1, . . . , 12 for third order equations as described in Corollary 2.4. L31 : 3.6, 3.41, 3.54, 3.57, 3.77, L32 : 3.54, 3.76, L33 : 3.73,

L35 : 3.33, 3.58,

L36 : 3.45,

L37 : 3.37, 3.38, 3.39, 3.56, 3.65, 3.68, 3.70,

L38 : 3.27, 3.46, 3.63, 3.74, 3.75, L39 : 3.1, 3.4, 3.5, 3.16, 3.17, 3.29, 3.47, 3.64, 3.66, 3.78, L310 : 3.32, 3.35, 3.42,

L312 : 3.18, 3.71.

Similar to equations of order two, in the listing below the decomposition of equations of Chapter 3 is given if its type is different from L31 . 3.1: y 000 + λy = Lclm(D − θ)y = 0, θ3 + λ = 0. Type L39 . 3.4: y 000 + 3y 0 − 4y = Lclm(D − 1, D − θ)y = 0, θ2 + θ + 4 = 0. Type L39 . 3.5: y 000 − a2 y 0 = Lclm(D + a, D − a, D)y = 0. Type L39 . 3.16: y 000 − 2y 00 − 3y 0 + 10y = Lclm(D + 2, D − θ)y = 0, θ2 + 4θ + 5 = 0. Type L39 . 3.17: y 000 − 2y 00 − a2 y 0 + 2a2 y = Lclm(D + a, D − a, D − 2)y = 0. Type L39 .   y = 0. 3.18: y 000 − 3ay 00 + 3a2 y 0 − a3 y = Lclm D − a − 2 2x + C1 x + C1 x + C2 3 Type L12 . 3 1 1
0 3 3.27: y 000 − 2y 00 − 11 4 y − 4 y = (D − 3)Lclm D + 2 − x + C y = 0. Type L8 . 3 y 00 + y = Lclm D + 1 + 1 , D − θ + 1
y = 0, θ2 − θ + 1 = 0. 3.29: y 000 + x x x Type L39 . − 1 )y 0 − x − 1 y = 3.32: y 000 − x +x 2ν y 00 − x − 2ν x x
Lclm D2 − 2ν x+ 1 D − 1, D − 1 y = 0. Type L310 . 3
y 00 + 4y 0 + 2 y = Lclm D − 1 + 1 )(D + x − 5
y = 0. 3.33: y 000 + x − x x x x+C x Type L35 . 3 x + − 52
ν−1 2(x + ν − 2) 00 000 ν 3.35: y − y0 − x 2 y = y + x+ x x 1 − 1 D + ν − 2 , D − 1
y = 0. Type L3 . Lclm D2 − x + 2ν 10 x x

Loewy Decompositions

401

2 y0 + 2 2 y= x2 − 2x x − 2x
1 2 1 D + x − 2 + x )Lclm(D − 1, D − x y = 0. Type L37 . 8 y = D +2+ 1 1
y = 0. 0 y + )Lclm(D −2, D − 3.38: y 000 − 2x8x x −1 2x − 1 x − 1/2 Type L37 . x + 4 y 00 + 2 y 0 = 3.39: y 000 + 2x −1 2x − 1 1

1+ 1 1 3 4 D+ x 1 Lclm D + 2 + 1 , D y = 0. Type L7 . x− 2 x− 2 1 y 00 + (x + 1 )y 0 = Lclm D2 − 1 D + 1, D
y = 0. Type L3 . 3.42: y 000 − x 10 x x
0 3
2 1
4 2 3 000 00 3.45: y + x y + x + x y + x y = D + x D + x D + 1 y = 0. Type L36 . 5 y 00 + 4 y 0 = D + 3
Lclm D − 1 + 1
y = 0. Type L3 . 3.46: y 000 + x 8 x x x+C x2
6 y 00 + 6 )y 0 = Lclm D + 2 , D + 1 , D y = 0. Type L3 . 3.47: y 000 + x 9 x x x2
3 3 4 )D + 2 D + 2
y = 0. 3.54: y 000 − x x− 6 y 00 − 2x 2− 6 y 0 +2y = D2 −(x2 − x x x x2 3 Type L2 . 3.55: y 000 + 28x y 00 + 210 y 0 = x +1 x +1   3 2 4x 2 x + 4x + 3C2 − 4 4x + 3C y = 0. Type L312 . Lclm D + 2 3 2 x + 1 x + C2 x + 2x + 3C2 x + C1 2 , D + θ)y = 0, θ2 + 1 = 0. 3.56: y 000 + 22x y 00 + y 0 + 22x y = Lclm(D − x x +2 x +2 Type L39 . 7 − 1
y 00 + 1 + 1
y 0 + 1 y = 3.58: y 000 + 14 + 2x x x2 4x2 2x2 1 − 2
D + 1 − 1 − 1
y = 0. Type L3 . Lclm D + C + 5 4 2x 4x2 x x
2 3 y 00 − 2 y 0 + 2 y = D + 1 Lclm D − 3x 2
3.63: y 000 + x 2 3 3 + x y = 0. x x x C +x Type L38 . 3 y 00 − a2 − 1 )y = Lclm D + a , D − 1 , D
y = 0. Type L3 . 3.64: y 000 + x 9 x x x2
4 y 00 + (1 + 8 )y = Lclm D − 2 , D + θ + 1 y = 0, θ2 + 1 = 0. 3.65: y 000 − x x x x2 3 Type L9 . 6 y 00 + (a − 12 )y = Lclm D − α + 2
y = 0, α3 + a = 0. Type L3 . 3.66: y 000 + x 9 x x2 x3   3 00 5x − 30 0 4x + 30 1 2x − 60 3.68: y 000 + x + x y + x2 y + x3 y = D + x + x2 − 60x + 450   6− 4x3 − 252x2 + 4032x − 20160 6 − p y = 0. Lclm D + x , D +1+ x q x4 − 84x3 + 2016x2 − 20160x + 75600 3.37: y 000 − y 00 −

p := 8x7 + 196x6 + 2700x5 + 25500x4 + 171600x3 + 801360x2 + 2358720x +3326400, q := x8 + 28x7 + 450x6 + 5100x5 + 42900x4 + 267120x3

402 +1179360x2 + 3326400x + 4536000. Type L37 .     12x − 12 y = 3x + 3 y 00 − 3.70: y 000 + x x2 + 1 x  x2 + 1   2x 2x 3 1 y = 0. Type L3 . D+ 2 + x Lclm D − 2 1 , D − 2 x + x 7 x +1 x +2 x +1    4 4 2 2 1 + 2 y 00 − 2 0 2 3 3 3 3 3.71: y 000 − x + 3 x x + 3 − x − x2 )y − ( x + 3 − x + x2 )y =   C1 + 2C2 x + 3x2 3 Lclm D − 2 3 y = 0. Type L12 . C1 (x + 1) + C2 x + x       8 −8+ 8 + 4 y= 6 + 6 + 4 y0 − 2 + 2 y 00 − 3.73: y 000 − x + 2 2 3 1 x x + 1 x x x+1 x x x  2 + 1 D − 1 )(D − 2 y = 0. Type L3 . D− x+ 3 x x 1 x   1 y 00 + 1 y 0 = DLclm D − 2x y = 0. Type L38 . 3.74: y 000 − x x2 C + x2       8 + 4 − 8x 2 + 2x y 00 + 42 + 2 6 y0 − x y= 3.75: y 000 − x 2 3 2 x +1 x x +1  x + 1 2 x 1 + 1 3Cx + C + 2x 3 D − 22x + x + 1 x − 1 Lclm D − Cx3 + Cx + x2 y = 0. Type L8 . x +1    2 + 1
D + 2 2 y = 0. Type L3 . 3.76: y 000 + 14 y 00 − 26 y 0 = D2 + x D− x 2 4 5 x x x x   3 4 − 1 + 4x + 4x + 2 y 00 3.78: y 000 − 2 − x x2 x4 + 2x2 + 2x + 1     8 − 2 + 8x3 + 8x + 4 4 + 1 − 4x3 + 4x + 2 + 1− x y0 + x y= 2 4 2 2 4 2 x x + 2x + 2x + 1 x x + 2x + 2x + 1   1 , D + 1 y = 0. Type L3 . Lclm D − 1, D − 1 − x 9 x2

Appendix E Symmetries of Kamke’s Collection

In this appendix the symmetry classes of a large number of quasilinear second order, and linear and quasilinear equations of third order from Kamke’s collection are listed. Equations that are not rational in all arguments, contain undetermined functions, more than two parameters or parameters in an exponent, are excluded. If there are parameters involved it is assumed that they are unconstrained, any relations between them may change the symmetry class. Equations of Second Order. These are the second order nonlinear equations of Chapter 6. The following tabulation is organized by symmetry class; for each class the numbers of the equations which it contains are given. T rivial : 6.3, 6.5, 6.6, 6.8, 6.9, 6.27, 6.108, 6.142, 6.144, 6.145, 6.147, 6.171, 6.211, 6.212. 2 S1 : 6.4, 6.10, 6.11, 6.21, 6.26, 6.31, 6.40, 6.45, 6.47, 6.50, 6.56, 6.73, 6.74, 6.79, 6.80, 6.82, 6.86, 6.87, 6.89, 6.90, 6.92, 6.94, 6.96, 6.98, 6.105, 6.106, 6.118, 6.119, 6.120, 6.141, 6.143, 6.153, 6.156, 6.160, 6.172, 6.189, 6.190, 6.219, 6.226, 6.233, 6.240, 6.245. 2 S2,1 : 6.227, 6.228, 6.229, 6.232, 6.239, 6.243. 2 S2,2 : 6.1, 6.2, 6.7, 6.12, 6.30, 6.32, 6.42, 6.43, 6.57, 6.78, 6.97, 6.104, 6.109, 6.110, 6.111, 6.127, 6.130, 6.137, 6.140, 6.146, 6.154, 6.155, 6.159, 6.174, 6.188, 6.205, 6.237. 2 S3,1 : none. 2 S3,2 : 6.57, 6.81, 6.133, 6.138, 6.158, 6.162, 6.163, 6.183, 6.209, 6.244 2 S3,3 : 6.71. 2 S3,4 : none. S82 : 6.93, 6.99, 6.107, 6.117, 6.124, 6.125, 6.126, 6.128, 6.134, 6.135, 6.150, 6.151, 6.157, 6.164, 6.168, 6.169, 6.173, 6.175, 6.176, 6.178, 6.179, 6.180, 6.181, 6.182, 6.184, 6.185, 6.186, 6.191, 6.192, 6.193, 6.194, 6.195, 6.206, 6.208, 6.210, 6.214. In the subsequent listing the number of the equation, its symmetry class and, if it is different from S82 , its symmetry generators are given. Whenever possible, the solution is given explicitly or it is indicated what type of transcendentals for its representation are required.

403

404 2 6.1: y 00 − y 2 = 0. Symmetry class S2,2 . Generators {∂x , x∂x − 2y∂y }. Z dy q + C2 . Solution: x = 2 3 y − C 1 3 2 6.2: y 00 − 6y 2 = 0. Symmetry class S2,2 . Generators {∂x , x∂x − 2y∂y }. Solution: Weierstraß’ P-function. 6.3: y 00 − 6y 2 − x = 0. Trivial symmetry. Solution: Painlev´e’s Transcendent. 6.4: y 00 − 6y 2 + 4y = 0. Symmetry class S12 . Generator ∂x . Solution: Elliptic integral. 6.5: y 00 + ay 2 + bx + c = 0. Trivial symmetry. Solution: Painlev´e’s Transcendent. 6.6: y 00 − 2y 3 − xy + a = 0. Trivial symmetry. Solution: Painlev´e’s Transcendent. 2 6.7: y 00 − ay 3 = 0. Symmetry class S2,2 . Generators {∂x , x∂x − y∂y }. Solution: Elliptic integral. 6.8: y 00 −2ay 3 +2abxy −b = 0. Trivial symmetry. Reduction to y 0 +ay 2 −bx = 0. 6.9: y 00 + ay 3 + bxy + cy + d = 0. Trivial symmetry. Solution: Painlev´e’s Transcendent. 2 2 6.10: y 00 + ay 3 + by Z + cy + d = 0. Symmetry class S1 . Generator ∂x . dy Solution: x = − q + C2 . 1 4 + 2 by 3 + cy 2 + 2dy + C ay 1 2 3

ν + 1∂ . 6.11: y 00 + axν y n = 0. Symmetry class S12 . Generator ∂x − n −1 y 2n 2n+1 2 00 = 0. Symmetry class S2,2 . 6.12: y + (n + 1)a y Z  dy 1 p Generators ∂x , x∂x − n y∂y . Solution: x = + C2 . 2n y (ay) − 1 + C1 6.21: y 00 − 3y 0 − y 2 − 2y = 0. Symmetry class S12 . Generator ∂x . 2 . 6.23: y 00 + 5ay 0 + 6ya2 − 6y 2 = 0. Symmetry class S2,2 ax Generators {∂x , e (∂x − 2ay∂y }. Solution: y = a2 C12 e−2ax P(C1 e−ax + C2 , 0, −1). 2 6.24: y 00 + 3ay 0 + 2a2 y − 2y 3 = 0. Symmetry class S2,2 . ax Generators {∂x , e (∂x − ay∂y )}. Solution: y = −iaC1 e−ax snk2 =−1 (C1 e−ax + C2 ). 2 6.26: y 00 + ay 0 + by n + a 4− 1 y = 0. Symmetry class S12 . Generator ∂x . 6.27: y 00 + ay 0 + bxν y n = 0. Trivial symmetry. 2 6.30: y 00 + yy 0 − y 3 = 0. Symmetry class S2,2 . Generators {∂x , x∂x − y∂y )}. 0 P (u, 0, 1) Solution: y = C1 where u = C1 x + C2 . P(u, 0, 1) 6.31: y 00 + yy 0 − y 3 + ay = 0. Symmetry class S12 . Generator ∂x .

Symmetries of Kamke’s Collection q 0 a P (u, 12, C1) with u = x a + C . Solution: 1 2 2 3 P(u, 12, C1 ) − 1 2 3 2 6.32: y 00 + yy 0 + 2ay 0 − y 3 + ay 2 + 2a2 y = 0. Symmetry class S2,2 . ax Generators {∂x , e (∂x − ay∂y )}. P 0 (u, 0, 1) Solution: y = C1 e−ax where u = Ca1 e−ax + C2 , a 6= 0. P(u, 0, 1)

405

q

6.40: y 00 − 3yy 0 − (3ay 2 + 4a2 y + b) = 0. Symmetry class S12 . Generator ∂x . 2 6.42: y 00 − 2ayy 0 = 0. Symmetry class S2,2 . Generators {∂x , x∂x − y∂y }. 2 6.43: y 00 + ayy 0 + by 3 = 0. Symmetry class S2,2 . Generators {∂x , x∂x − y∂y }. Z Z udu dy +C , u is defined by +log y = C1 . Solution: x = 2 2 2 uy log y 2u + au + b 6.45: y 00 + ay 02 + byZ= 0. Symmetry class S12 . Generator ∂x . dy Solution: x = 2a2 + C2 . 2 −2ay 2a C1 e + b(1 − 2ay) 6.47: y 00 + ay 02 + by 0 + cy = 0. Symmetry class S12 . Generator ∂x . Reduction to Abel’s equation.

6.50: y 00 + ayy 02Z+ by = 0. Symmetry class S12 . Generator ∂x . dy p Solution: x = + C2 . C1 exp (−ay 2 ) − b/a 6.56: y 00 + ay(y 0 + s 1)2 = 0. Symmetry class S12 . Generator ∂x . Z ay 2 + C1 dy + C2 . Solution: x = 1 − ay 2 − C2 6.57 y 00 − a(xy 0 − y)ν = 0. Zν = 1: Symmetry class S82 .
dx . Solution: y = C1 x + C2 x exp 12 x2 x 2 ν = 3: Symmetry class S3,2 . Generators {y∂x , x∂y , x∂x − y∂y }. Solution: (C12 C22 + a)y 2 − 2C12 C2 xy + C22 x2 − C1 = 0. 2 y∂ }. 2 ν = 2 or ν ≥ 4: Symmetry class S2,2 . Generators {x∂y , x∂x − ν − 1 y Solution: Parameter representation. 6.58: y 00 − kxa y b y 0c = 0. Symmetry class S12 . Generator ∂x . 2 6.71: 8y 00 + 9y 04 = 0. Symmetry class S3,2 . Generators {∂x , ∂y , x∂x + 23 y∂y }. 3 2 Solution: (C1 + y) = (C2 + x) . 2 y∂ . 6.73: xy 00 + 2y 0 − xy n = 0. Symmetry class S12 . Generator x∂x − n − 1 y 00 02 ν n 2 6.74: xy + 2y + ax y = 0. Symmetry class S8 for n = 1; symmetry ν + 1 y∂ ; symmetry class S 2 class S12 for n ≥ 2, ν 6= n, generator x∂x − n 2 −1 y y for ν = n ≥ 2, additional generator ∂x − x ∂y . 2 6.78: xy 00 + yy 0 − y 0 = 0. Symmetry class S2,2 .
Generators {x∂x , x log x∂x +(y−2)∂y }. Solution: y = 2−2 tan C1 (log x − C2 ) .

6.79: xy 00 − x2 y 02 + 2y 0 + y 2 = 0. Symmetry class S12 . Generator x∂x − y∂y .

406 Z

dη with η = xy. C2 e + 2η + 1 6.80: xy 00 + a(xy 0 − y)2 = b. Symmetry class S12 . Generator x∂y . Solution: x = C1 exp

η

2 6.81: 2xy 00 + y 03 + y 0 = 0. Symmetry class S3,2 . Generators {∂y , x∂x + y∂y , xy∂x + 21 y 2 ∂y }. Solution: (y + C1 )2 = 2C2 x − C22 .

6.82: xy 00 = a(y n − y). Symmetry class S12 . Generator x∂x . 6.86: x2 y 00 + a(xy 0 − y)2 = bx2 . Symmetry class S12 . Generator x∂y . 6.87: x2 y 00 + ayy 02 + bx = 0. Symmetry class S12 . Generator x∂x + y∂y . 6.89: y 00 + xy 00 + y 02 = 0. Symmetry class S12 . Generator ∂y . Solution: y = C1 + C2 x + (C22 ) log x − C2 . 6.90: 4x2 y 00 − x4 y 02 + 4y = 0. Symmetry class S12 . Generator x∂x − 2y∂y . 2 6.91: 9x2 y 00 + ay 3 + 2y = 0. Symmetry class S2,2 .  y 2/3 Generators x∂x , 3x ∂x − 1/3 ∂y . Solution in terms of elliptic functions. x 6.92: 9x3 (y 00 + yy 0 − y 3 ) + 12xy + 24 = 0. Symmetry class S12 . Generator x∂x − y∂y . Solution: Parameter representation. x 6.93: 9x2 y 00 −a(xy 0 −y)2 = 0. Symmetry class S82 . Solution: y = x a log C1 x + C2 . 6.94: 2x2 y 00 + (2x3 y + 9x2 )y 0 − 2x3 y 3 + 3x2 y 2 + axy + b = 0. Symmetry class S12 . Generator x∂x − y∂y . 2 y∂ . 6.96: x4 y 00 + a2 y ν = 0. Symmetry class S12 . Generator x∂x + ν − 1 y 4 00 0 3 0 2 2 6.97: x y − 2xyy − x y + 4y = 0. Symmetry class S2,2 . Generators {x∂x + 2y∂y , x log x∂x + (2y log x + x2 − y)∂y }. Solution: y = [C2 tan(C1 log x + C2 )]x2 .

6.98: x4 y 00 − x2 y 02 − x3 y 0 + 4y 2 = 0. Symmetry class S12 . Z dy +C2 . Generator x∂x +2y∂y . Solution: x = exp y y 2 x (C1 exp ( 2 ) − 4 2 − 2 x x 6.99: x4 y 00 + (xy 0 − y)3 = 0. Symmetry class S82 . Solution: y = C1 x + x arcsin Cx2 . 2 6.104: yy 00 − a = 0. Symmetry class S2,2 . Generators {∂x , x∂x + y∂y }. Z dy p Solution: x = . 2a log y + C1 6.105: yy 00 − ax = 0. Symmetry class S12 . Generator x∂x + 32 y∂y . 6.106: yy 00 − ax2 = 0. Symmetry class S12 . Generator x∂x + 2y∂y . √ 6.107: yy 00 +y 02 −a = 0. Symmetry class S82 . Solution y = ax2 + C1 x + C2 . 6.108: yy 00 + y 02 = ax + b. Trivial symmetry. 2 6.109: yy 00 + y 02 − y 0 = 0. Symmetry class S2,2 . Generators {∂x , x∂x + y∂y }. Solution: x = y + C1 log y − C1 + C2 .

Symmetries of Kamke’s Collection

407

2 6.110: yy 00 − y 02 + 1 = 0. Symmetry class S2,2 . Generators {∂x , x∂x + y∂y }. Solution: C1 y = sin (C1 x) + C2 . 2 6.111: yy 00 − y 02 − 1 = 0. Symmetry class S2,2 . Generators {∂x , x∂x + y∂y }. Solution: C1 y = cosh (C1 x) + C2 . 6.117: y 00 y − y 02 + ay 0 y + by 2 = 0. Symmetry class S82 . Solution y = exp (C1 e−ax − ab x + C2 ). 6.118: y 00 y − y 02 + ay 0 y + by 3 − 2ay 2 = 0. Symmetry class S12 . Generator ∂x . 6.119: y 00 y − y 02 + (ay − 1)y 0 − 2b2 y 3 + 2a2 y 2 + ay = 0. Symmetry class S12 . Generator ∂x . 6.120: yy 00 − y 02 + a(y − 1)y 0 − y(y + 1)(b2 y 2 − a2 ) = 0. Symmetry class S12 . Generator ∂x . 6.124: y 00 y − 3y 02 + 3y 0 y − y 2 = 0. Symmetry class S82 . Solution: (C1 ex + 1)y 2 + C2 e2x = 0. 6.125: y 00 y − ay 02 = 0. Symmetry class S82 . Solution: y = (C1 x + C2 )1/(1−a) for a 6= 1, y = C1 eC2 x for a = 1. 2 6.126: yy 00 + ay 02 + a = 0. Symmetry class S2,2 . Z dy p Generators {∂x , x∂x + y∂y }. Solution: x = + C2 . C1 y −2a − 1 2 6.127: yy 00 + ay 02 + by 3 = 0. Symmetry class S2,2 Generators {∂x , x∂x − 2y∂y }. Solution: Parameter representation. 6.128: yy 00 + ay 02 + byy 0 + cy 2 + dy 1−a = 0. Symmetry class S82 . 2 6.130: yy 00 + ay 02 + by 2 y 0 + cy 4 = 0. Symmetry class S2,2 . Generators {∂x , x∂x − y∂y }. 2 6.133: y 00 (y + x) + y 02 − y 0 = 0. Symmetry class S3,2 .  2 1 2 2 Generators ∂x − ∂y , x∂x + y∂y , (x − 3 xy − 3 y )∂x + ( 13 x2 + 23 xy − y 2 )∂y . 2 Solution: y 2 − (3C2 + 94 C1 + 2x)y + 49 C22 + 3C2 x − 49 C1 x + 81 64 C1 + x = 0. 00 00 02 0 2 6.134: yy − xy − 2y − 2y = 0. Symmetry class S8 . C2 . Solution y = C1 + x − C1 6.135: yy 00 − xy 00 + y 03 + y 02 + y 0 + 1 = 0. Symmetry class S82 . Solution: (y + C1 )2 = C22 − (x + C1 )2 . 2 6.137: 2yy 00 + y 02 + 1 = 0. Symmetry class S2,2 . Generators {∂x , x∂x + y∂y }. Solution: x = C1 (t − sin t) + C2 , y = C1 (1 − cos t). 2 . 6.138: 2yy 00 − y 02 + a = 0. Symmetry class S3,2 2 Generators {∂x , x∂x + y∂y , x ∂x + 2xy∂y }. Solution: C1 y = ( 21 C1 x + C2 )2 − a. 2 . 6.140: 2y 00 y − y 02 − 8y 3 = 0. Symmetry class S2,2 Z dy 1 p Generators {∂x , x∂x − 2y∂y }. Solution: x + 2 = C2 . y(y 2 − C1 )

6.141: 2y 00 y − y 02 − 8y 3 − 4y 2 = 0. Symmetry class S12 . Generator ∂x .

408 Solution: Elliptic functions. 6.142: 2y 00 y − y 02 − 8y 3 − 4xy 2 = 0. Trivial symmetry. 6.143: 2y 00 y − y 02 + ay 3 + by 2 = 0. Symmetry class S12 . Generator ∂x . 6.144: 2y 00 y − y 02 + ay 3 + 2xy 2 + 1 = 0. Trivial symmetry. 6.145: 2y 00 y − y 02 + ay 3 + bxy 2 = 0. Trivial symmetry. 2 6.146: 2yy 00 − y 02 + 3y 4 = 0. Symmetry class S2,2 . Generators {∂x , x∂x − y∂y } Solution: Elliptic function.

6.147: 2yy 00 − y 02 − 3y 4 − 8xy 3 − 4(x2 + a)y 2 + b = 0. Trivial symmetry. Solution: Painlev´e’s Transcendent. C1 . 6.150: 2yy 00 − 3y 02 = 0. Symmetry class S82 . Solution: y = x + C2 C1 00 02 2 2 6.151: 2yy −3y −4y = 0. Symmetry class S8 . Solution: y = 2. cos (x + C2 ) 6.153: 2yy 00 − 6yZ02 + ay 5 + y 2 = 0. Symmetry class S12 . Generator ∂x dy q + C2 . Solution: x = y C1 y 4 + ay 3 + 14 2 6.154: 2yy 00 − y 04 − y 02 = 0. Symmetry class S2,2 . Generators {∂x , x∂x + y∂y } Solution: x = C1 t + C2 sin t + C2 , y = C1 (1 − cos t). 2 6.155: 4yy 00 − 4ay 00 + 2y 02 + 2 = 0. Symmetry class S2,2 . Generators {∂x , x∂x + (y − a)∂y }. r p a−y Solution: x = 12 (y − a + C2 )(a − y) − C2 arctan y − a + C + C1 . 2

6.156: 3yy 00 − 2y 02 = ax2 + bx + c. Symmetry class S12 . 3b
∂ . Generator (ax2 + bx + c)∂x + 3a + 2a y 6.157: 3yy 00 − 5y 02 = 0. Symmetry class S82 . Solution: y =

1 . (C1 x + C2 )3/2

2 6.158: 4yy 00 − 3y 02 + 4y = 0. Symmetry class S3,2 .

Generators {∂x , x∂x +2y∂y , x2 ∂x +4xy∂y }. Solution x = 12

Z

dy q √ + C2 . C1 y + y

2 . 6.159: 4yy 00 − 3y 02 − 12y 3 = 0. Symmetry class S2,2 Z p C1 − 4y 3 Generators {∂x , x∂x − 2y∂y }. Solution x = dy + C2 . √ y y

6.160: 4yy 00 − 3y 02 + ay 3 + by 2 + cy = 0. Symmetry class S12 . Generator ∂x 6.162: 4yy 00 − 5y 02 + ay 3 = 0. 2 Symmetry class S3,2 . Generators {∂x , x∂x − 2y∂y , x2 ∂x − 4xy∂y }. 1 √ 1 Solution: 2C12 C2 y 4 + (C12 C2 − C1 x2 − 16 a) y + C12 = 0. 2 6.163: 12yy 00 − 15y 02 + 8y 3 = 0. Symmetry class S3,2 . 2 Generators {∂x , x∂x − 2y∂y , x ∂x − 4xy∂y }.

Symmetries of Kamke’s Collection

409

6C2 . [(C1 + x)2 + C2 ]2 6.164: nyy 00 −(n−1)y 02 = 0. Symmetry class S82 . Solution: y = (C1 x+C2 )n . 6.168: (ay + b)y 00 + cy 02 = 0. Symmetry class S82 . Solution: y = (C1 x+C2 )a/(a+c) − ab if a+c 6= 0, y = C1 eC2 x − ab if a+c = 0. r 1 00 02 0 2 6.169: xyy +xy −yy = 0. Symmetry class S8 . Solution: y = . C1 x2 + C2 Solution: y =

6.171: xyy 00 − xy 02 + yy 0 + axy 4 + by 3 + cy + d = 0. Trivial symmetry. 6.172: xyy 00 − xy 02 + ayy 0 + bxy 3 = 0. Symmetry class S12 . Generator x∂x − 2y∂y . 02 0 2 6.173: xyy 00 + 2xy r + ayy = 0. Symmetry class S8 . 1 Solution: y = . C1 x1−a + C2 2 6.174: xyy 00 − 2xy 02 + yy 0 + y 0 = 0. Symmetry class S2,2 . Generators {x∂x , x log x∂x + y∂y }. 6.175: xyy 00 − 2xy 02 + ayy 0 = 0. Symmetry class S82 . 1 1 Solution: y = for a 6= 1, y = for a = 1. C1 log x + C2 C1 x1−a + C2 6.176: xyy 00 − 4xy 02 + 5yy 0 + y 0 = 0. Symmetry class S82 . x Solution: y = . (C1 x3 + C2 )1/3 6.178: x(y + x)y√00 + xy 02 − (y − x)y 0 − y = 0. Symmetry class S82 . Solution: y = C1 x2 + C2 − x. 6.179: 2y 00 yx − y 02 x√+ y 0 y = 0. Symmetry class S82 . Solution: y = C1 ( x + C2 )2 . 6.180: x2 (y − 1)y 00 − 2x2 y 02 − 2x(y − 1)y 0 − 2y(y − 1)2 = 0. 2 Symmetry class S82 . Solution: y = C12x + C2 x . C1 x + C2 x − 1 6.181: x2 (y + x)y 00 − (xy 0 − y)2 = 0. Symmetry class S82 .
Solution: y = C1 x exp Cx2 − x. 6.182: x2 (y − x)y 00 − a(xy 0 − y)2 = 0. Symmetry class S82 .
1/(a−1) Solution: y = C1 x C x +x a − 1 + x. 2

2 6.183: 2x2 yy 00 − x2 y 02 + y 2 − x2 = 0. Symmetry class S3,2 . Generators

2 {x∂x +y∂y , x log (x)∂x + log (x)+1 y∂y , x log (x) ∂x + log (x)+2 log (x)y∂y }.
√ Solution: y = x C1 + 4C1 C2 − 1 log x + C2 log x2 . 6.184: ax2 yy 00 + bx2 y 02 + cxyy 0 + dy 2 = 0 Symmetry class S82 . 6.185: x(x + 1)2 yy 00 − x(x + 1)2 y 0 + 2(x + 1)2 yy 0 − ax(x + 2)y 2 = 0.
Symmetry class S82 . Solution: y = C1 (x + 1)a exp Cx2 . 6.186: 8(x3 − 1)yy 00 − 4(x3 − 1)y 02 + 12x2 yy 0 − 3x(x + 2)y 2 = 0 Symmetry class S82 . Solution in terms of hypergeometric functions.

410 2 6.188: y 2 y 00 − a = 0. Symmetry class S2,2 . Generators {∂x , x∂x + 23 y∂y }. Z √ ydy p Solution x = + C2 . C1 y − 2a 6.189: y 2 y 00 yy 02 + ax = 0. Symmetry class S12 . Generator x∂x + y∂y .
6.190: y 2 y 00 +yyy 02 = ax+b. Symmetry class S12 . Generator x+ ab ∂x +y∂y . 1 (ax + b)3 = C. First integral: y 3 y 03 − 3y 2 y 0 (ax + b) + ay 3 + a

6.191: y 00 (y 2 + 1) − y 02 (2y − 1) = 0. Symmetry class S82 . Solution: y = tan log(C1 x + C2 ) 6.192: (y 2 + 1)y 00 − 3yy 02 = 0. Symmetry class S82 . Solution: y = p C1 x + C2 C1 x + C2 )2 − 1 2 00 6.193: (y + x)y + 2(y 2 − x)y 03 + 4yy 02 + y 0 = 0. Symmetry class S82 . Solution: y = C1 eC2 y − x. 6.194: (y 2 + x)y 00 − (y 02 + 1)(xy 0 −
y) = 0. Symmetry class S82 . y Solution: y = C1 exp arctan C2 x . 6.195: (y 2 + x2 )y 00 − 2(y 02 + 1)(xy 0 − y) = 0. Symmetry class S82 . Solution: y 2 + C1 y + C2 x + x2 = 0. 6.205: xy 2 y 00 − a = 0. 2 Symmetry class S2,2 . Generators {x∂x + 31 y∂y , x2 ∂x + xy∂y }. Solution: Parameter representation. 2 2 02 2 2 0 6.206: (x2 − a2 )(y 2 − a2 )y 00 − (x p − a )yy + x(y − √ a )y = 0. 2 2 2 2 Symmetry class S8 . Solution y − a + y = C1 ( x − a2 + x) + C2 .

6.208: x3 y 2 y 00 + (xy 0 − y)3 (y + x) = 0. Symmetry class S82 . 6.209: y 3 y 00 − a = 0. 2 Symmetry class S3,2 . Generators {∂x , x∂x + 21 y∂y , x2 ∂x + xy∂y }. Solution: (C2 x − C2 )2 + C1 y 2 + a = 0. 00 6.210: (y 3 + y)yr − (3y 2 − 1)y 02 = 0. Symmetry class S82 . C1 x − C2 Solution: y = 1 − C1 x + C2 .

6.211: 2y 3 y 00 + y 4 − a2 xy 2 = 1. Trivial symmetry. 6.212: 2y 3 y 00 + y 2 y 02 = ax2 + bx + c. Trivial symmetry. 6.214: (4y 3 − g2 y − g3 )y 00 − (6y 2 − 21 g2 )y 02 = 0; g2 and g3 constant. Symmetry class S82 . Solution: y = P(C1 x + C2 , g2 , g3 ); P Weierstraß’ P-function. 6.219: (y 2 + ax2 + 2bx + c2 )y 00 + dy = 0. Symmetry class S12 . Generator (ax2 + 2bx + c)∂x + (xy + ab y)∂y . 6.226: y 0 y 00 − x2 yy 0 − xy 2 = 0. Symmetry class S12 . Generator y∂y . 2 6.227: xy 0 y 00 − yy 00 + 4y 02 = 0. Symmetry class S2,1 . Generators {x∂x , y∂y }. Solution: x = C1 (t − 1)e2t , y = C2 te−2t .

6.228: xy 0 y 00 − yy 00 − y 04 − 2y 02 − 1 = 0.

Symmetries of Kamke’s Collection

411

2 Symmetry class S2,1 . Generators {x∂x + y∂y , (x + y)∂y }. 00 0 3 2 6.229: ay y x + by 2 = 0. Symmetry class S2,1 . Generators {x∂x , y∂y }. 2 0 2 00 03 0 6.231: (2y y + x )y + 2yy + 3xy + y = 0. Symmetry class S12 . Generator x∂x + y∂y . First integral: y 2 y 02 + x2 y 0 + xy = C. 6.232: y 00 y 02 + y 00 y 2 + y 3 = 0. Symmetry class S12 . Generators {∂x , y∂y }. Z √ √ 1/2  dx + C2 . (1 + 43 cot2 (x 3 + C1 ) Solution: log y = 21 log | sin (x 3 + C1 )| ±

6.233: (y 02 + a(xy 0 − y))y 00 = b. Symmetry class S12 . Generator ∂x − 12 ax∂y . 6.236: y 002 + ay + b = 0. 
2 Symmetry class S2,2 . Generators ∂x , x∂x + 4 y + ab ∂y . 6.237: a2 y 002 − 2axy 00 + y 0 = 0. 2 Symmetry class S2,2 . Generators {∂y , x∂x + 3y∂y }. 2 002 6.238: 2(x + 1)y − x(4y 0 + y)y 00 + 2(y 0 + x)y 0 − 2y = 0. Trivial symmetry. 2 6.239: 3x2 y 002 − 2(3xy 0 + y)y 00 + 4y 02 = 0. Symmetry class S2,1 . 2 2 2 Generators {x∂x , y∂y }. Solution y = C1 x + C1 C2 x + C2 . 6.240: (9x3 − 2x2 )y 002 − 6x(6x − 1)y 0 y 00 − 6y 0 y 00 + 36xy 02 = 0. Symmetry class S12 . Generator y∂y . Solution y = C12 x3 + C1 C2 x + C22 . 2 6.243: (a2 y 2 − b2 )y 002 − 2a2 yy 02 y 00 + (a2 y 02 − 1)y 02 = 0. Symmetry class S2,1 . r 1 (b2 + 1 ). Generator ∂x . Solution y = C1 exp (C2 x) ± a C22 2 6.244: (x2 yy 00 − x2 y 02 + y 2 )2 = 4xy(xy 0 − y)3 . Symmetry class S3,2 .  y x Generators ∂x + x ∂y , x∂x + y log y ∂y , ∂y . Solution y = C1 x exp C 1− x . 2

6.245: (2yy 00 − y 02 )3 + 32y 00 (xy 00 − y 03 )3 = 0. Symmetry class S12 .
Generator x∂x + y∂y . Solution y = 1 3 (C12 x + 1)2 + 2C22 . C1 C2 Nonlinear Equations of Third Order. These equations are from Chapter 7 of Kamke’s collection. S13 : 7.2, 7.5, 7.6.

3 S2,1 : 7.1.

3 S2,2 : 7.3, 7.4, 7.7.

3 S4,4 : 7.12

S63 : 7.10, 7.11.

S73 : 7.8, 7.9.

3 7.1: y 000 − a2 (y 05 − 2y 03 − y 0 ) = 0. Symmetry class S2,1 . Generators {∂x , ∂y }. Solution in parameter representation: Z Z 1 dp 1 pdp q q x= + C2 , y = + C3 . a a 1 2 C1 + 3 (p + 1)3 C1 + 31 (p2 + 1)3

7.2: y 000 + yy 00 − y 02 = 0. Symmetry class S13 . Generator ∂x . 3 7.3: y 000 − yy 00 + y 02 = 0. Symmetry class S2,2 . Generators {∂x , x∂x − y∂y }. Reduction to Abel equation.

412 3 7.4: y 000 + ayy 00 = 0. Symmetry class S2,2 . Generators {∂x , x∂x − y∂y }. Reduction to Abel equation. 7.5: x2 y 000 + xy 00 + (2xy − 1)y 0 + y 2 = 0. Symmetry class S13 . Generator x∂x − y∂y . 7.6: x2 y 000 + xyy 00 − xy 00 + xy 02 − yy 0 + y 0 = 0. Symmetry class S13 = {x∂x }. First integral: xy 00 + (y − 1)y 0 = Cx. 3 0 3 7.7: yy 000 − y 0 y 00 + Z y y = 0. Symmetry class S2,2 . Generator {∂x , x∂x − y∂y }. dy + C3 . Solution: x = q 2 C1 y + C2 − 14 y 4

7.8: 4y 2 y 000 − 18yy 0 y 00 + 15y 03 = 0. Symmetry class S73 . 1 Solution: y = . (C1 x2 + C2 x + C3 )2 7.9: 9y 2 y 000 − 45yy 0 y 00 + 40y 03 = 0. Symmetry class S73 . Generators {∂x , x∂x , x2 ∂x − 3xy∂y }. 1 Solution: y 2 = . (C1 x2 + C2 x + C3 )3 7.10: 2y 0 y 000 − 3y 002 = 0. Symmetry class S63 . 1 x + C2 Generators {∂x , ∂y , x∂x , y∂y , x2 ∂x , y 2 ∂y }. Solution: y = C C x+C . 3

4

7.11: y 02 y 000 + y 000 − 3y 0 y 002 = 0. Symmetry class S63 . Generators {∂x , ∂y , x∂x + y∂y , y∂x − x∂y , xy∂x − 12 (x2 − y 2 )∂y , (x2 − y 2 ∂x + 2xy∂y }. Solution: (y − C1 )2 + (x − C2 )2 = C32 . 3 7.12: y 02 y 000 + y 000 − 3y 0 y 002 − ay 002 = 0. Symmetry class S4,4 . Generators {∂x , ∂y , x∂x + y∂y , y∂x − x∂y }. Solution: x = C2 + C1 e−at (a cos t − sin t), y = C3 + C1 e−at (a sin t + cos t). Linear Equations of Third Order. These equations are from Chapter 3 of 3 3 Kamke’s collection. There occur the three symmetry classes S4,5 , S5,1 and S73 , 3 and the parametrized class S5,2 (a). The symmetry generators are not given explicitly because they are closely related to the elements of a fundamental system (see Exercise 5.3). Whenever a closed form for the latter exists it is also given. It is usually obtained by means of a Loewy decomposition. 3 : 3.2, 3.7, 3.30, 3.31, 3.32, 3.33, 3.34, 3.35, 3.37, 3.38, 3.39, 3.40, S4,5 3.41,3.42, 3.49, 3.50, 3.51, 3.53, 3.54, 3.56, 3.58, 3.60, 3.65, 3.66, 3.68, 3.70, 3.73, 3.75, 3.76, 3.77. 3 3 S5,1 : 3.27, 3.63, 3.74. S5,2 (a) : 3.1, 3.4, 3.16, 3.29, 3.71. 3 S7 : 3.6, 3.21, 3.45, 3.47, 3.48, 3.57, 3.61, 3.64. 3 3.1: y 000 + λy = 0. Symmetry class S5,2 (θ), θ2 − θ + 1 = 0. Fundamental √ √ 1 1 system: {exp (−θ1 x), exp ( 2 θ1 x) cos 2 θ1 3x), exp ( 21 θ1 x) sin 12 θ1 3x)}, θ1 real root of θ3 − λ = 0. 3 3.2: y 000 + ax3 y + bx = 0. Symmetry class S4,5 .

Symmetries of Kamke’s Collection

413

3 3.3: y 000 = axb y. Symmetry class S4,5 . 3 3.4: y 000 + 3y 0 − 4y = 0. Symmetry class S5,2 (θ), θ2 − θ + 25 = 0. Fundamental √ √ 1 1 system: {exp(x), exp (− 2 x) cos ( 2 15x), exp (− 21 x) sin ( 12 15x)}. 3.5: y 000 −a2 y 0 = 0. Symmetry class S73 . Fundamental system: {1, exp (±ax)}. 3.6: y 000 + 2axy 0 + ay = 0. Symmetry class S73 . Fundamental system: {u2 , uv, v 2 }, u and v fundamental system for w00 + 12 axw = 0. 3 3.7: y 000 − x2 y 00 + (a + b − 1)y 0 − aby = 0. Symmetry class S4,5 . 4 4 3 3.16: y 000 − 2y 00 − 3y 0 + 10y = 0. Symmetry class S5,2 (θ), θ2 − 17 θ + 17 = 0. Fundamental system: {exp (−2x), cos(x) exp (2x), sin(x) exp (2x)}. 3 3.17: y 000 − 2y 00 − a2 y 0 + 2a2 y = 0. Symmetry class: S5,2 (3) for a = 1, 3 3 3 S5,1 for a = 2, S5,2 (6) for a = 3, S5,2 (4) for a = 4. Fundamental system: {exp (2x), exp (±ax)}. 3 3.18: y 000 − 3ay 00 + 3a2 y 0 − a3 y = 0. Symmetry class S5,2 (θ), θ2 − θ + 1 = 0. 2 Fundamental system: {exp (ax), x exp (ax), x exp (ax)}. 3.20: y 000 −6xy 00 −(4a+8x2 −2)y 0 −8axy = 0. Symmetry class S73 . Fundamental system: {u2 , uv, v 2 }; u and v fundamental system of w00 −2xw +aw = 0. 2 2 0 3 3 3 3.21: y 000 + 3axy 00 + √ class  3a x y + a x√y = 0. Symmetry  S7 . Solution: 1 2 y = exp (− 2 ax ) C1 + C2 cos (x 3a) + C3 sin (x 3a) . 3 3.27: 4y 000 − 8y 00 − 11y 0 − 3y = 0. Symmetry class S5,1 . 1 Fundamental system: {exp (3x), exp (− 2 x), x exp (− 12 x)}. 3 3.29: xy 000 + 3y 00 + xy = 0. Symmetry class S5,2 (θ), θ2 − θ + 1 = 0. √  1 −x 1 Fundamental system: x e , x exp ( 12 (1 ± i 3)x) . 3 3.30: xy 000 + 3y 00 − ax2 y = 0. Symmetry class S4,5 . 000 00 0 3 3.31: xy + (a + b)y − xy − ay = 0. Symmetry class S4,5 . 000 00 0 3 3.32: xy + (a + b)y − xy − ay = 0. Symmetry class S4,5 . Solution: y = C1 ex + xν+1 [C2 Jν+1 (ix) + C3 Yν+1 (ix)]. 3 3.33: xy 000 − (x + 2ν)y 00 − (x − 2ν − 1)y 0 + (x − 1)y = 0. Symmetry class S4,5 .  5 R 1 2 1 2 dx 1 2 5 Fundamental system: x exp (− 2 x ), x exp (− 2 x ) exp ( 2 x ) 6 . x 3 . 3.34: xy 000 + 32 y 00 + 21 axy = 0. Symmetry class S4,5

3.35: xy 000 − 2(x + ν − 1)y 00 + (x + 3ν − 52 )y 0 − (ν − 12 )y = 0. 3 Symmetry class S4,5 . 

 x Solution: y = C1 e + xν exp ( 12 x) C2 Jν 12 ix + C3 Yν 12 ix . 3 3.37: (x − 2)xy 000 − (x − 2)xy 00 − 2y 0 + 2y = 0. Symmetry class S4,5 .  2 x x R −x   x Fundamental system: x , e , e e x(x − 2) log x − 2 − 2(x − 1) dx . 3 . Fundamental system: 3.38: (2x − 1)y 000 − 8xy 0 + 8y = 0. Symmetry class S4,5 Z Z n x exp (−4x)dx exp (−2x)dx o exp (2x), x exp (2x), exp (2x) −x . 4x(x − 1) + 1 4x(x − 1) + 1 3 3.39: (2x − 1)y 000 + (x + 4)y 00 + 2y 0 = 0. Symmetry class S4,5 .

414 3 3.40: x2 y 000 − 6y 0 + ax2 y = 0. Symmetry class S4,5 . 3 3.41: x2 y 000 + (x + 1)y 00 − y = 0. Symmetry class S4,5 . 3 3.42: x2 y 000 − xy 00 + (x2 + 1)y 0 = 0. Symmetry class S4,5 . 3 3.45: x2 y 000 + 4xy 00 + (x2 + 2)y 0 + 3xy = 0. Symmetry class S4,5 . 3 3.46: x2 y 000 + 5xy 00 + 4y = 0. Symmetry class: S5,1 . log x 1 Fundamental system: {1, x , x }. 3.47: x2 y 000 + 6xy 00 + 6y 0 = 0. Symmetry class S73 . 1 , 1 }. Fundamental system: {1, x x2 00 0 2 2 000 3.48: x y + 6xy + 6y + ax y = 0. Symmetry class S73 . Fundamental system: { 12 exp θx}, θ3 + a = 0. x 3 3.49: x2 y 000 − 3(p + q)xy 00 + 3p(3q + 1)y 0 − x2 y = 0. Symmetry class S4,5 .

3.50: x2 y 000 − 2(n + 1)xy 00 + (ax2 + 6n)y 0 − 2axy = 0, n ∈ N. 3 Symmetry class S4,5 . √ √ Fundamental system: {ax2 + 4n − 2, exp (x −a)P (x), exp (−x −a)Q(x)}, P (x) and Q(x) polynomials of degree ≤ 2n + 2. 3.51: x2 y 000 − x(x − 2)y 00 + (x2 + ν 2 − 14 )y 0 − (x2 − 2x + ν 2 − 41 )y = 0. 3 Symmetry class S4,5 . 3.52: x2 y 000 − (x + ν)xy 00 + ν(2x + 1)y 0 − ν(x + 1)y = 0. 3 Symmetry class S4,5 . 3.53: x2 y 000 − x(x − 1)y 00 + (x2 − 2x − ν 2 + 14 )y 0 − (ν 2 − 14 )y = 0. 3 Symmetry class S4,5 . 3.54: x3 y 000 − (x4 − 6x)y 00 − (2x3 − 6)y 0 + 2x2 y = 0. 3 Symmetry class S4,5 . 3 3.55: (x2 + 1)y 000 + 8xy 00 + 10y 0 = 0. Symmetry class S4,5 . o n 2 2 2 x(x + 3) x (x + 2) 1 . Fundamental system: 2 2, 2 2, 2 2 (x + 1) (x + 1) (x + 1) 3 3.56: (x2 + 2)y 000 − 2xy 00 + (x2 + 2)y 0 − 2xy = 0. Symmetry class S4,5 . 2 Fundamental system: {x , cos x, sin x}.

3.57: x(x − 1)y 000 + 3(x − 21 )y 00 + (ax + 12 b)y 0 + 12 ay = 0. Symmetry class S73 . Fundamental system: {y12 , y1 y2 , y22 }, y1 and y2 hypergeometric functions. 3.58: 4x2 y 000 + (x2 + 14x − 1)y 00 + 4(x + 1)y 0 + 2y = 0. 3 Symmetry class S4,5 . Solution: Z  h 5 3
2 √ x2 + 1 i +1 C + y = x exp − x 4x C2 x− 2 + C3 x− 2 exp . 1 4x 3 . 3.60: x3 y 000 (ν 2 − 1)xy 00 + (ax3 + ν 2 − 1)y = 0. Symmetry class S4,5

3.61: x2 y 000 + (4x3 − 4ν 2 x + x)y 0 + (4ν 2 − 1)y = 0. Symmetry class S73 . Fundamental system: {xJν2 (x), xJν (x)Yν (x), xYν2 (x)}. 3.62: x3 y 000 + (ax2ν + 1 − ν 2 )xy 0 + [bx3ν + a(ν − 1)x2ν + ν 2 − 1]y = 0.

Symmetries of Kamke’s Collection

415

3 Symmetry class S4,5 . 3 000 2 00 3 3.63: x y + 3x y − 2xy 0 + 2y = 0. Symmetry class S5,1 .  1 Fundamental system: x, 2 , x log x . x 3.64: x3 y 000 + 3x2 y 00 + (1 − a2 )xy 0 = 0. Symmetry class S73 . Fundamental system: {1, xa , 1a }. x 3 3.65: x3 y 000 − 4x2 y 00 + (x2 + 8)xy 0 − 2(x2 + 4)y = 0. Symmetry class S4,5 . 2 Fundamental system: {x , x cos x, x sin x}. 3 3.66: x3 y 000 + 6x2 y 00 + (ax3 − 12)y = 0. Symmetry class S4,5 . 3  θx − 2 exp (θx) , θ + a = 0. Fundamental system: x3 3.68: x3 y 000 + (x + 3)x2 y 00 + 5(x − 6)xy 0 + (4x + 30)y = 0. 3 Symmetry class S4,5 . Z io
6 n 2 h 5 d 2 d −5 −x Solution: y = x x x e C1 + C2 log x x4 ex dx + C3 . 6 2 dx dx 3 . 3.70: (x2 + 1)xy 000 + 3(2x2 + 1)y 00 − 12y = 0. Symmetry class S4,5 2 u + 1 2 Fundamental system: {2x + 1, xu, 2x + 3x − xu log u − 1 }, with √ u = x2 + 1. 3 3.71: (x+3)x2 y 000 −3(x+2)xy 00 +6(x+1)y 0 −6y = 0. Symmetry class S5,2 (3). 2 3 Fundamental system: {x + 1, x , x }. 3.73: (x + 1)x3 y 000 − (4x + 2)x2 y 00 + (10x + 4)xy 0 − 4(3x + 1)y = 0. Symmetry 2 3 class S4,5 . Fundamental system: {x2 , x2 log x, x + x3 + x2 log (x) )}. 3 3.74: 4x4 y 000 − 4x3 y 00 + 4x2 y 0 = 1. Symmetry class S5,1 . 1 2 2 Fundamental system: {1, x , x log x − 36x }. 3.75: (x2 + 1)x3 y 000 − (4x2 + 2)x2 y 00 + (10x2 + 4)xy 0 − 4(3x2 + 1)y = 0. 3 Symmetry class S4,5 . Fundamental system: {x2 , x3 + x, x2 log |x|}. 6 000 2 00 3 3.76: x y + x y − 2y = 0. Symmetry class S4,5 . 6 000 5 00 3 3.77: x y + 6x y + ay = 0. Symmetry class S4,5 .

Appendix F ALLTYPES Userinterface

To a large extent the calculations involved in the analysis of differential equations described in this book can hardly be performed by pencil and paper. Therefore, the userinterface of the computer algebra system ALLTYPES has been developed which is available on the website www.alltypes.de. It provides a great number of functions that perform the laborious calculations involved automatically. A complete listing of the functions currently provided may be obtained interactively by submitting UserFunctions(); after the ALLTYPES system has been loaded. A detailed description of an individual function with the name FunctionName is obtained by calling UserFunction FunctionName; In this Appendix the connection to the mathematical background for any particular function is provided. Subsequently, all user interface functions are listed in alphabetical order and the page number where it is explained is given. If there occur several page numbers, they correspond to the various argument types, i.e., the corresponding Methods in the ALLTYPES system. AbsoluteInvariants(LODE|ABEL) Returns absolute invariants for linear and Abel’s equation. CharacteristicPolynomial(LIEVEC|LIEALG) Returns characteristic polynomial of Lie algebra. CommutatorTable(LIEVEC) Returns commutator table of Lie algebra. DeterminingSystem(ODE|PDE) Returns determining system for Lie symmetries. ExactQuotient(LODE|LDFMOD) Returns exact quotient or failed if it does not exist. ForsythQuadrinvariants(LODE) Returns Forsyth’s quadrinvariants for equation of order up to three. Gcd(LODE|LDFMOD) Returns greatest common divisor of the two arguments. GroebnerBase(DPOLID) Returns Gr¨ obner basis in specified term order. Integral(POLY|PARF|RATF) Returns integral of first argument w.r.t. second.

417

418 Invariants(LODE) Returns different kinds of invariants. JacobianProlongation(LIEVEC) Returns k-th prolongation of first argument in Jacobian form. JanetBase(ODE |LDFMOD|LFMOD) Returns Janet base in specified term order. JanetResolution(LDFMOD) Returns Janet resolution. LaguerreForsythNormalForm(LODE3) Returns Laguerre-Forsyth normal form. Lclm(LODE|LDFMOD) Returns least common left multiple of the two arguments. LieCanonicalForm(ODE2|ODE3) Returns canonical form corresponding to Lie symmetry class. LiouvillianSolutions(LODE|LDFMOD) Returns Liouvillian solutions following from first order components. LoewyDecomposition(LODE) Returns Loewy decomposition into irreducible components. Prolongation(LDO |LIEVEC) Returns k-th prolongation of first argument. RationalNormalForm(LODE| RICCATI|ABEL) Returns rational normal form. RationalSolution(UPOLY|LODE|RICCATI ) Returns most general rational solution including constants. RationalSolutions(UPOLY|LODE|RICCATI) Returns all rational solutions. RelativeInvariants(LODE|ABEL) Returns relative invariants. SemiSimple?(LIEALG|LIEVEC) Returns t if Lie algebra is semi-simple. Socle(LODE|LDFMOD) Returns largest completely reducible right component. Solvable?(LIEALG| LIEVEC) Returns t if Lie algebra is solvable. Solve(LODE|ODE) Returns solution or failed if it cannot be found. Symmetries(ODE| PDE) Returns vector fields of infinitesimal generators. SymmetryClass(LODE|ODE ) n Returns Lie symmetry class Si,j . TresseInvariants(ODE) Returns Tresse invariants of order not higher than four.

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Index

differential Galois theory for second order equations, 35 differential invariant, 114 differential type, 40 divisor, 70 Drach’s equation, 104

Abel’s equation, 179 absolute invariant, 113, 161 adjoint group, 109 Appell, 179 associated equations, 24 autoreduced system, 49 autoreduction, 47

elementary extension, 16 Emden-Fowler equation, generalized, 221 equivalence of first order ode’s, 176 of linear second order ode’s, 168 of linear third order ode’s, 172 equivalence class, 160 equivalence problem, 2, 160 equivalence transformation, 160 equivalent rational solutions, 17 essential group parameters, 107 exact quotient, 22 exact quotient module, 70

base field, 8, 11 Bernoulli equation, 180 Bouton, 165 center, 117 centralizer, 117 characteristic equation, 117 characteristic polynomial, 117 characteristic system, 78 coherent, 53 commutation relation, 108 commutator group, 106 complete set of monomials, 384 complete system, 51, 109 completely reducible, 27 component, 70 connected, 109 contact transformation, 159, 195 criticoid, 159

factorization, 21 full set of solutions, 13 full system of differential invariants, 115 function, 42 fundamental invariants, 201 fundamental system, 12, 78

defining equations, 112 dependent variables, 42 derivation operator, 42 derivative, 42 derivative operator, 42 derivative vector, 44 derived Lie algebra, 118 derived series, 118 determining system, 199 Dickson’s lemma, 44

gauge, 40 Gcd, 21 Gcrd, 23 general solution, 311 Goursat, 160 graded lexicographic ordering, 45

431

432 graded reverse lexicographic ordering, 45 greatest common divisor, 21 greatest common left divisor, 23 greatest common right divisor, 23 group type, 138 homomorphism of Lie groups, 106 hyperexponential solution, 86 imprimitive Lie group, 108 independent variables, 42 infinitesimal generator, 108 inhomogeneous equation, 14 integrability conditions, 50 integrable Lie algebra, 118 integrable Lie group, 106 integrable pair, 86 integral basis, 78 integrating factor, 313, 349 intersection module, 69 intransitive Lie group, 108 invariant absolute, 161 of a differential equation, 160 of a Lie group, 113 relative, 161 semi-, 161 invariant decomposition, 108 invariants absolute of linear ode, 168 relative of linear ode, 166–168 irreducible, 24 isomorphism of Lie groups, 106 Jacobi normal form, 79 Janet basis, 11 Janet basis algorithm for, 53 classification of, 57, 61 construction of from solutions, 75 decomposition of, 99 definition of, 53 exact quotient of, 70

Index for Lie vector fields, 143 type of, 55 k-fold transitive, 108 Krause, 193 Laguerre, 166 Laguerre-Forsyth canonical form, 164 Landau, 26 Lclm, 23 Lcm, 21 leading derivative, 44 leading term, 44 least common left multiple, 23 least common multiple, 21 Levi decomposition, 121 lexicographic ordering, 45 Lie, 10 Lie algebra, 116 Lie algebra classical, 120 classification of, 135 definition of, 116 derived, 118 exceptional, 120 integrable, 118 semi-simple, 119 solvable, 118 Lie group, 106 Lie system, 144 Lie transformation group, 107 Lie’s equation, 219 Lie’s first theorem, 109 Lie’s relations, 143 Lie’s second theorem, 110 Lie’s third theorem, 110 Lie-determinant, 201 line element, 195 linear ode, 12 Liouville R., 181 Liouville, R., 186 Liouvillian extension, 16 local Lie group, 106 Loewy decomposition, 11 Loewy decomposition

Index type of, 28 Loewy factor, 27 lower equations, 201 matrice de cˆ otes, 44 matrix of weights, 44 Michel, 193 Neumer, 162 normalizer, 117 orbit, 108 order of a derivative, 42 ordinary differential field, 42 ordinary differential ring, 42 parametric derivative, 55 partial differential field, 42 partial differential ring, 42 path curve, 126 Picard-Vessiot extension, 16 point transformation, 159, 194 primitive Lie group, 108 principal derivative, 55 prolongation of a symmetry generator, 197 of a vector field, 114 quotient of ordinary differential operators, 22 radical, 119 rank of a Janet basis, 55 of a Lie algebra, 117 ranking, 43 rational normal form definition of, 162 of Abel’s equation, 180 of Riccati’s equation, 177 reduction, 47 relative invariant, 161 relative syzygies module, 70 Riccati equation, 16 Riccati equation

433 associated to a linear equation, 16 Riccati-like partial differential equations, 85 semi-invariant, 161 semi-simple Lie algebra, 119 semi-simple Lie group, 106 similar Lie groups, 111 simple Lie algebra, 119 simple Lie group, 106 singular solution, 311 solvable Lie algebra, 118 solvable Lie group, 106 special functions, 169 special rational solutions, 17 St¨ackel, 162 stabilizer, 107 stratum, 193 structure constants, 107 structure invariance group definition of, 160 of Abel’s equation, 179 of Lie’s equation, 186 of linear differential equation, 162 of Riccati’s equation, 177 symmetric power and Lie symmetries, 245 definition of, 33 symmetric product, 33 symmetry, 3, 196 symmetry of an ode, 195 symmetry algebra, 193 symmetry class, 4, 7, 193 symmetry group, 193 symmetry type, 4, 193 system of imprimitivity, 108 term, 43 term ordering definition of, 43 grevlex, 45 grlex, 45

434 lex, 45 transitive Lie group, 108 Tresse, 188 trivial symmetries, 206 typical differential dimension, 40 unconnected, 109 variation of constants, 14 weight, 161 Wronskian, 12

Index